Q: The area of a right-angled triangle is $40$ times its base. What is its height?

Concept & Algebraic Formulation: Translate the textual conditions directly into an algebraic equation using the standard area formula. A right-angled triangle uses its perpendicular legs as the base and height. Area = (1/2) × b × h Step-by-Step Solution: • Let the base of the triangle be b and its height be h. • According to the given condition, Area = 40 × b. • We also know the standard formula: Area = (1/2) × b × h. • Equate the two expressions for area: (1/2) × b × h = 40 × b. • Assuming the base b is non-zero, divide both sides by b: (1/2) × h = 40. • Multiply both sides by 2 to isolate h: h = 80. Exam Strategy & Shortcut: Recognize that if Area = k × base, then (1/2) × height = k. Therefore, the height is simply always 2k. Here, k = 40, so height is 2 × 40 = 80. This can be solved mentally in two seconds. Common Pitfall: Choosing "Data inadequate" because only one numerical value (40) is given. Realizing the variable for base (b) cancels out is the key insight to avoiding this trap. Final Answer: Therefore, the correct answer is 80 cm.

Q: What is the area of the given figure? <img src="/api/images/area-given-composite-figure-rectangle-triangles.png" alt="area given composite figure rectangle triangles" width="300" height="200" loading="lazy" decoding="async" class="ro…

Concept & Composite Area: To find the area of a composite figure, break it down into simpler geometric shapes (rectangles and triangles). The figure consists of a central rectangle and two right-angled triangles attached at the ends. We use the Pythagorean theorem to find missing side lengths. Pythagorean theorem: a^2 + b^2 = c^2 Step-by-Step Solution: • Given: A central rectangle with length 12 cm. Two right-angled triangles at the ends with legs of 4 cm and 6 cm. • Analyze the Triangles: The right angle symbol indicates the legs are 4 cm and 6 cm. Area of one triangle = (1/2) × 4 × 6 = 12 cm^2. Total area for two triangles = 12 × 2 = 24 cm^2. • Find Rectangle Width: The hypotenuse of these right triangles forms the width of the central rectangle. Width = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} cm. \sqrt{52} \approx 7.21 cm (or typically approximated as 7.2 in such problems). • Calculate Rectangle Area: Area = Length × Width = 12 × \sqrt{52} \approx 12 × 7.2 = 86.4 cm^2. • Total Area: Total Area = Rectangle Area + Triangles Area = 86.4 + 24 = 110.4 cm^2. Exam Strategy & Shortcut: Recognize that \sqrt{52} is slightly larger than \sqrt{49} (which is 7). 12 × 7 = 84. Total area is slightly more than 84 + 24 = 108. Looking at the options, 110.4 cm^2 is the only logical choice that perfectly aligns with a standard approximation of \sqrt{52} \approx 7.2. Common Pitfall: Misidentifying the hypotenuse. If one assumes the 6 cm side is the hypotenuse instead of a leg, the calculations will lead to an incorrect and missing option. Always pay close attention to where the right-angle square symbol is placed. Final Answer: Therefore, the correct answer is 110.4 cm².

Q: The perimeter of an isosceles triangle is equal to 14 cm and the lateral side is to the base in the ratio $5 : 4$. The area of the triangle is

Concept & Isosceles Triangle Properties: An isosceles triangle features two equal lateral sides. By drawing a perpendicular altitude straight down to the base, you split the triangle into two identical right-angled triangles, allowing you to use the Pythagorean theorem to easily find the height. Area of a triangle: A = (1/2) × base × height Step-by-Step Solution: • Given: The total perimeter is 14 cm. The ratio of one lateral side to the base is 5 : 4. Because it is isosceles, the two lateral sides are equal. Let the sides be 5x, 5x, and the base be 4x. • Find Side Lengths: Perimeter = 5x + 5x + 4x = 14x 14x = 14 \Rightarrow x = 1 The actual side lengths are exactly 5 cm, 5 cm, and 4 cm. • Find Height: Drop a perpendicular line from the top vertex down to the base. This perfectly bisects the 4 cm base into two 2 cm segments. We now have a right triangle with a hypotenuse of 5 and a base of 2. h^2 + 2^2 = 5^2 h^2 + 4 = 25 h^2 = 21 \Rightarrow h = \sqrt{21} cm. • Calculate Area: A = (1/2) × total base × height A = (1/2) × 4 × \sqrt{21} = 2\sqrt{21} cm^2. Exam Strategy & Shortcut: Quickly resolve the ratio 5:5:4 to sum to 14, meaning the sides are identically 5, 5, and 4. Knowing the altitude bisects the base to 2, mentally calculate h = \sqrt{25 - 4} = \sqrt{21}. Area is simply half-base × height = 2 × \sqrt{21}. Common Pitfall: Misinterpreting the phrasing "lateral side is to the base" as meaning there is only one lateral side involved, leading to a flawed perimeter equation like 5x + 4x = 14. Always remember a triangle must have three sides. Final Answer: Therefore, the correct answer is 2\sqrt{21} cm^2.

Q: The sides of a triangle are in the ratio of $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. If the perimeter is 52 cm, then the length of the smallest side is

Concept & Fractional Ratios: When a ratio is presented in fractions, it is mathematically much cleaner to convert it into a ratio of whole integers by multiplying all terms by the Least Common Multiple (LCM) of their denominators. Step-by-Step Solution: • Given: The ratio of the sides is (1/2) : (1/3) : (1/4) and the total perimeter is 52 cm. • Convert Ratio to Integers: Find the LCM of the denominators 2, 3, and 4, which is 12. Multiply each fraction in the ratio by 12: ((1/2) × 12) : ((1/3) × 12) : ((1/4) × 12) This simplifies to a clean integer ratio of 6 : 4 : 3. • Set up Equation: Let the physical sides be 6x, 4x, and 3x. Perimeter = 6x + 4x + 3x = 13x 13x = 52 x = (52/13) = 4 • Find Smallest Side: The smallest side mathematically corresponds to the smallest ratio part, which is 3. Smallest side = 3x = 3 × 4 = 12 cm. Exam Strategy & Shortcut: Once you have formulated the integer ratio 6:4:3, simply sum the parts (6+4+3=13). Divide the total perimeter by this sum (52 / 13 = 4). To find the specific smallest side, just multiply this base multiplier by the smallest ratio component (4 × 3 = 12). Common Pitfall: Attempting to work directly with the fractions to solve for the perimeter without normalizing them first. This method is much slower and highly prone to arithmetic mistakes involving common denominators. Final Answer: Therefore, the correct answer is 12 cm.

Q: The sides of a triangle are consecutive integers. The perimeter of the triangle is 120 cm. Find the length of the greatest side.

Concept & Consecutive Integers: Consecutive integers can be represented algebraically as x, x+1, and x+2. However, defining them symmetrically as x-1, x, and x+1 is a powerful algebraic trick because their sum perfectly cancels out the integer constants. Step-by-Step Solution: • Given: The three sides are strictly consecutive integers. The perimeter (the sum of all sides) is exactly 120 cm. • Set up Variables: Let the three sides be represented symmetrically as (x - 1), x, and (x + 1). • Formulate the Equation: (x - 1) + x + (x + 1) = 120 • Solve for x: Notice how the -1 and +1 cancel out effortlessly. 3x = 120 x = 40 • Find Greatest Side: The individual sides are 40-1=39, 40, and 40+1=41. The greatest side is represented by (x + 1) = 41 cm. Exam Strategy & Shortcut: If three consecutive numbers sum up to a specific value, the middle number is always exactly the average of that sum. 120 / 3 = 40. So the consecutive numbers must be 39, 40, and 41. The greatest of these is clearly 41. This allows you to solve the entire problem mentally in just seconds. Common Pitfall: Setting the sides up the traditional way as x, x+1, x+2 (3x+3=120 \Rightarrow 3x=117 \Rightarrow x=39), and then hastily selecting 39 (which is the smallest side) instead of completing the final required step to find the greatest side (x+2 = 41). Final Answer: Therefore, the correct answer is 41 cm.

Q: The area of a triangle is $216\text{ cm}^2$ and its sides are in the ratio $3 : 4 : 5$. The perimeter of the triangle is

Concept & Ratio Properties: When the sides of a triangle are in the ratio 3 : 4 : 5, they form a Pythagorean triplet (since 3^2 + 4^2 = 5^2). This means the triangle is a right-angled triangle. The area of a right-angled triangle is given by: Area = (1/2) × Base × Height Step-by-Step Solution: 1. Let the sides of the triangle be 3x, 4x, and 5x. 2. The base and height are the legs of the right triangle: 3x and 4x. 3. Substitute the values into the area formula: 216 = (1/2) × 3x × 4x 4. Simplify the equation: 216 = 6x^2 x^2 = 36 x = 6 5. The sides of the triangle are 3(6) = 18 cm, 4(6) = 24 cm, and 5(6) = 30 cm. 6. Calculate the perimeter: Perimeter = 18 + 24 + 30 = 72 cm. Exam Strategy & Shortcut: Instead of calculating individual sides, note that the perimeter in terms of x is 3x + 4x + 5x = 12x. Once you find x = 6 from the area equation 6x^2 = 216, directly multiply 12 × 6 = 72. This saves calculation time. Common Pitfall: A common mistake is assuming the sides are exactly 3, 4, and 5 and just finding the perimeter of that basic triangle (12 cm). Always use a variable (like x) to represent the common ratio multiplier. Final Answer: Therefore, the correct answer is 72 cm.

Q: The sides of a triangle are 5 cm, 6 cm, and 7 cm. One more triangle is formed by joining the mid-points of the sides. The perimeter of the second triangle in cm is

Concept & Midpoint Theorem for Perimeter: According to the midpoint theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is exactly half of its length. Consequently, the perimeter of the inner triangle formed by joining the midpoints is exactly half the perimeter of the original triangle. Perimeter_{new} = \frac{Perimeter_{original}}{2} Step-by-Step Solution: 1. Calculate the perimeter of the original triangle by summing its sides: Original Perimeter = 5 + 6 + 7 = 18 cm. 2. The sides of the new triangle are half the lengths of the original sides: New sides = (5/2), (6/2), and (7/2) (which are 2.5 cm, 3 cm, and 3.5 cm). 3. Calculate the perimeter of the new triangle: New Perimeter = 2.5 + 3 + 3.5 = 9 cm. (Alternatively, just divide the original perimeter by 2: (18/2) = 9 cm). Exam Strategy & Shortcut: Save time by finding the sum of the original sides and immediately halving the result. 5 + 6 + 7 = 18 \rightarrow (18/2) = 9. Common Pitfall: A common mistake is trying to divide the area by 4 and then reverse-engineering the side lengths. Perimeter scales linearly (factor of (1/2)), while area scales quadratically (factor of (1/4)). Final Answer: Therefore, the correct answer is 9.

Q: A triangle of area $9 y \text{ cm}^2$ has been drawn such that its area is equal to the area of an equilateral triangle of side $6\text{ cm}$. The value of $y$ would be

Concept & Equilateral Triangle Area: The area of an equilateral triangle can be determined if the side length is known using the standard formula: Area = \frac{\sqrt{3}}{4} a^2 where a is the side of the triangle. Step-by-Step Solution: 1. Identify the given side length of the equilateral triangle: a = 6 cm. 2. Calculate the area of this equilateral triangle: Area = \frac{\sqrt{3}}{4} × (6)^2 Area = \frac{\sqrt{3}}{4} × 36 = 9\sqrt{3} cm^2. 3. We are given that the area of another triangle is 9 y cm^2 and it equals the area of this equilateral triangle. 4. Equate the two expressions for the area: 9 y = 9\sqrt{3} 5. Solve for y: y = \sqrt{3}. Exam Strategy & Shortcut: Memorize the formula for the area of an equilateral triangle. When setting up the equation 9y = \frac{\sqrt{3}}{4}(36), the 9s immediately cancel out, leaving y = \sqrt{3} instantly. Common Pitfall: A common mistake is confusing the formula for the height of an equilateral triangle (\frac{\sqrt{3}}{2}a) with the formula for the area. Final Answer: Therefore, the correct answer is \sqrt{3}.

Question 1

The areas of a square and a rectangle are equal. The length of the rectangle is greater than the length of any side of the square by 5 cm and the breadth is less by 3 cm. Find the perimeter of the rectangle.

Accepted Answer

Concept & Algebraic Modeling: When given comparative dimensions, base all variables on a single unknown. Here, base the rectangle's dimensions on the square's side, then equate their areas. Area_{sq} = Area_{rect} \Rightarrow s^2 = l × b Perimeter_{rect} = 2(l + b) Step-by-Step Solution: 1. Let the side of the square be x cm. 2. The area of the square is x^2. 3. The length of the rectangle is l = (x + 5) cm. 4. The breadth of the rectangle is b = (x - 3) cm. 5. The area of the rectangle is l × b = (x + 5)(x - 3). 6. Equate the areas: x^2 = (x + 5)(x - 3) x^2 = x^2 - 3x + 5x - 15 x^2 = x^2 + 2x - 15 7. Simplify the equation: 0 = 2x - 15 2x = 15 \Rightarrow x = 7.5 cm. 8. Now, find the dimensions of the rectangle: Length l = 7.5 + 5 = 12.5 cm. Breadth b = 7.5 - 3 = 4.5 cm. 9. Calculate the perimeter of the rectangle: Perimeter = 2(l + b) = 2(12.5 + 4.5) = 2(17) = 34 cm. Exam Strategy & Shortcut: Notice the equation s^2 = (s+5)(s-3) always simplifies to 0 = 2s - 15. You can jump straight to 2s = diff of product = 15 for these specific "rectangle vs square area equal" structures. s = 7.5. Perimeter = 2( (s+5) + (s-3) ) = 2(2s + 2) = 4s + 4 = 4(7.5) + 4 = 30 + 4 = 34. Common Pitfall: A common mistake is finding the value of the square's side (x = 7.5) and trying to double it or miscalculating the perimeter formula as just (l+b) instead of 2(l+b). Final Answer: Therefore, the correct answer is 34 cm.

Question 2

The area of a square is twice that of a rectangle. The perimeter of the rectangle is 10 cm. If its length and breadth each is increased by 1 cm, the area of the rectangle becomes equal to the area of the square. The length of side of the square is

Accepted Answer

Concept & Systems of Equations: This problem requires setting up a system of equations linking the areas and perimeters of two distinct geometric shapes. Area of Rectangle = l × b Perimeter of Rectangle = 2(l + b) Step-by-Step Solution: 1. Let the side of the square be S. Its area is S^2. 2. Let the length and breadth of the rectangle be l and b. Its area is lb. 3. We are given: S^2 = 2lb (Equation 1). 4. The perimeter of the rectangle is 10 cm: 2(l + b) = 10 \Rightarrow l + b = 5 (Equation 2). 5. If length and breadth are increased by 1 cm, the new area equals the square's area: (l + 1)(b + 1) = S^2 6. Expand the left side: lb + l + b + 1 = S^2. 7. Substitute (l + b) = 5 into the expanded equation: lb + 5 + 1 = S^2 \Rightarrow lb + 6 = S^2. 8. We already know from Equation 1 that S^2 = 2lb. Substitute this in: lb + 6 = 2lb \Rightarrow lb = 6. 9. Now plug lb = 6 back into Equation 1 to find S: S^2 = 2(6) = 12. 10. The length of the side of the square is S = \sqrt{12} = \sqrt{4 × 3} = 2\sqrt{3} cm. Exam Strategy & Shortcut: Instead of solving for individual l and b, recognize the algebraic pattern in (l+1)(b+1) = lb + (l+b) + 1. Since l+b is immediately known from half the perimeter (5), you can skip to lb + 6 = New Area = Square Area = 2lb, solving for lb in one fluid mental step. Common Pitfall: Trying to solve the quadratic equation to find exactly what l and b are (l=3, b=2). While correct, it wastes valuable time; you only need their product (lb = 6) to find the square's area. Final Answer: Therefore, the correct answer is 2\sqrt{3} cm.

Question 3

A rectangular plank $\sqrt{2}$ metre wide is placed symmetrically on the diagonal of a square of side 8 metres as shown in the figure. The area of the plank is  <img    src="/api/images/rectangular-plank-sqrt2-metre-wide-symmetrically-diagonal-squar…

Accepted Answer

Concept & Geometry of Diagonals: The plank forms a rectangle placed centrally over the diagonal of the square. Because the plank is constrained within the square, its ends are cut perpendicular to its length, creating small 45^\circ-45^\circ-90^\circ right triangles at the square's corners. Diagonal of Square = side × \sqrt{2} Step-by-Step Solution: 1. The side of the square is a = 8 m. 2. The length of the square's diagonal is D = a\sqrt{2} = 8\sqrt{2} m. 3. The plank is placed symmetrically along this diagonal. Its width is w = \sqrt{2} m. 4. This means the plank extends \frac{\sqrt{2}}{2} m on either side of the diagonal line. 5. Look at the corner of the square where the plank ends. The corner of the square forms a 90^\circ angle, and the diagonal bisects this into two 45^\circ angles. 6. The end of the plank is perpendicular to the diagonal, creating an isosceles right triangle at the square's tip. 7. The height of this small corner triangle (from the square's vertex to the end of the plank) is exactly half the plank's width: \frac{\sqrt{2}}{2}. 8. In an isosceles right triangle, the hypotenuse (the cut end of the plank) is twice the height along the symmetric axis: 2 × \frac{\sqrt{2}}{2} = \sqrt{2}. This confirms the plank remains a perfect rectangle inside the square. 9. The length of the plank, L, is the total diagonal minus the parts cut off at both ends. L = Total Diagonal - 2 × (distance from corner to plank end) L = 8\sqrt{2} - 2 ( \frac{\sqrt{2}}{2} ) = 8\sqrt{2} - \sqrt{2} = 7\sqrt{2} m. 10. The area of the rectangular plank is L × w: Area = 7\sqrt{2} × \sqrt{2} = 7 × 2 = 14 sq. m. Exam Strategy & Shortcut: Instead of complex geometric derivations, use the diagonal property. The plank trims a small isosceles right triangle off each end of the diagonal. Because the slope is 45^\circ, the length removed from the diagonal at each end exactly equals half the plank width. Length = 8\sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 7\sqrt{2}. Area = 7\sqrt{2} × \sqrt{2} = 14. Common Pitfall: Assuming the length of the plank is equal to the full diagonal of the square (8\sqrt{2}) without accounting for the fact that a rectangular plank with finite width cannot fit completely into a sharp 90^\circ corner without extending beyond the square. Final Answer: Therefore, the correct answer is 14 sq. m.

Question 4

The area of a right-angled triangle is $40$ times its base. What is its height?

Accepted Answer

Concept & Algebraic Formulation: Translate the textual conditions directly into an algebraic equation using the standard area formula. A right-angled triangle uses its perpendicular legs as the base and height. Area = (1/2) × b × h Step-by-Step Solution: • Let the base of the triangle be b and its height be h. • According to the given condition, Area = 40 × b. • We also know the standard formula: Area = (1/2) × b × h. • Equate the two expressions for area: (1/2) × b × h = 40 × b. • Assuming the base b is non-zero, divide both sides by b: (1/2) × h = 40. • Multiply both sides by 2 to isolate h: h = 80. Exam Strategy & Shortcut: Recognize that if Area = k × base, then (1/2) × height = k. Therefore, the height is simply always 2k. Here, k = 40, so height is 2 × 40 = 80. This can be solved mentally in two seconds. Common Pitfall: Choosing "Data inadequate" because only one numerical value (40) is given. Realizing the variable for base (b) cancels out is the key insight to avoiding this trap. Final Answer: Therefore, the correct answer is 80 cm.

Question 5

What is the area of the given figure?  <img    src="/api/images/area-given-composite-figure-rectangle-triangles.png"   alt="area given composite figure rectangle triangles"   width="300"   height="200"   loading="lazy"   decoding="async"   class="ro…

Accepted Answer

Concept & Composite Area: To find the area of a composite figure, break it down into simpler geometric shapes (rectangles and triangles). The figure consists of a central rectangle and two right-angled triangles attached at the ends. We use the Pythagorean theorem to find missing side lengths. Pythagorean theorem: a^2 + b^2 = c^2 Step-by-Step Solution: • Given: A central rectangle with length 12 cm. Two right-angled triangles at the ends with legs of 4 cm and 6 cm. • Analyze the Triangles: The right angle symbol indicates the legs are 4 cm and 6 cm. Area of one triangle = (1/2) × 4 × 6 = 12 cm^2. Total area for two triangles = 12 × 2 = 24 cm^2. • Find Rectangle Width: The hypotenuse of these right triangles forms the width of the central rectangle. Width = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} cm. \sqrt{52} \approx 7.21 cm (or typically approximated as 7.2 in such problems). • Calculate Rectangle Area: Area = Length × Width = 12 × \sqrt{52} \approx 12 × 7.2 = 86.4 cm^2. • Total Area: Total Area = Rectangle Area + Triangles Area = 86.4 + 24 = 110.4 cm^2. Exam Strategy & Shortcut: Recognize that \sqrt{52} is slightly larger than \sqrt{49} (which is 7). 12 × 7 = 84. Total area is slightly more than 84 + 24 = 108. Looking at the options, 110.4 cm^2 is the only logical choice that perfectly aligns with a standard approximation of \sqrt{52} \approx 7.2. Common Pitfall: Misidentifying the hypotenuse. If one assumes the 6 cm side is the hypotenuse instead of a leg, the calculations will lead to an incorrect and missing option. Always pay close attention to where the right-angle square symbol is placed. Final Answer: Therefore, the correct answer is 110.4 cm².

Question 6

The perimeter of an isosceles triangle is equal to 14 cm and the lateral side is to the base in the ratio $5 : 4$. The area of the triangle is

Accepted Answer

Concept & Isosceles Triangle Properties: An isosceles triangle features two equal lateral sides. By drawing a perpendicular altitude straight down to the base, you split the triangle into two identical right-angled triangles, allowing you to use the Pythagorean theorem to easily find the height. Area of a triangle: A = (1/2) × base × height Step-by-Step Solution: • Given: The total perimeter is 14 cm. The ratio of one lateral side to the base is 5 : 4. Because it is isosceles, the two lateral sides are equal. Let the sides be 5x, 5x, and the base be 4x. • Find Side Lengths: Perimeter = 5x + 5x + 4x = 14x 14x = 14 \Rightarrow x = 1 The actual side lengths are exactly 5 cm, 5 cm, and 4 cm. • Find Height: Drop a perpendicular line from the top vertex down to the base. This perfectly bisects the 4 cm base into two 2 cm segments. We now have a right triangle with a hypotenuse of 5 and a base of 2. h^2 + 2^2 = 5^2 h^2 + 4 = 25 h^2 = 21 \Rightarrow h = \sqrt{21} cm. • Calculate Area: A = (1/2) × total base × height A = (1/2) × 4 × \sqrt{21} = 2\sqrt{21} cm^2. Exam Strategy & Shortcut: Quickly resolve the ratio 5:5:4 to sum to 14, meaning the sides are identically 5, 5, and 4. Knowing the altitude bisects the base to 2, mentally calculate h = \sqrt{25 - 4} = \sqrt{21}. Area is simply half-base × height = 2 × \sqrt{21}. Common Pitfall: Misinterpreting the phrasing "lateral side is to the base" as meaning there is only one lateral side involved, leading to a flawed perimeter equation like 5x + 4x = 14. Always remember a triangle must have three sides. Final Answer: Therefore, the correct answer is 2\sqrt{21} cm^2.

Question 7

The sides of a triangle are in the ratio of $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$.  If the perimeter is 52 cm, then the length of the smallest side is

Accepted Answer

Concept & Fractional Ratios: When a ratio is presented in fractions, it is mathematically much cleaner to convert it into a ratio of whole integers by multiplying all terms by the Least Common Multiple (LCM) of their denominators. Step-by-Step Solution: • Given: The ratio of the sides is (1/2) : (1/3) : (1/4) and the total perimeter is 52 cm. • Convert Ratio to Integers: Find the LCM of the denominators 2, 3, and 4, which is 12. Multiply each fraction in the ratio by 12: ((1/2) × 12) : ((1/3) × 12) : ((1/4) × 12) This simplifies to a clean integer ratio of 6 : 4 : 3. • Set up Equation: Let the physical sides be 6x, 4x, and 3x. Perimeter = 6x + 4x + 3x = 13x 13x = 52 x = (52/13) = 4 • Find Smallest Side: The smallest side mathematically corresponds to the smallest ratio part, which is 3. Smallest side = 3x = 3 × 4 = 12 cm. Exam Strategy & Shortcut: Once you have formulated the integer ratio 6:4:3, simply sum the parts (6+4+3=13). Divide the total perimeter by this sum (52 / 13 = 4). To find the specific smallest side, just multiply this base multiplier by the smallest ratio component (4 × 3 = 12). Common Pitfall: Attempting to work directly with the fractions to solve for the perimeter without normalizing them first. This method is much slower and highly prone to arithmetic mistakes involving common denominators. Final Answer: Therefore, the correct answer is 12 cm.

Question 8

The sides of a triangle are consecutive integers. The perimeter of the triangle is 120 cm. Find the length of the greatest side.

Accepted Answer

Concept & Consecutive Integers: Consecutive integers can be represented algebraically as x, x+1, and x+2. However, defining them symmetrically as x-1, x, and x+1 is a powerful algebraic trick because their sum perfectly cancels out the integer constants. Step-by-Step Solution: • Given: The three sides are strictly consecutive integers. The perimeter (the sum of all sides) is exactly 120 cm. • Set up Variables: Let the three sides be represented symmetrically as (x - 1), x, and (x + 1). • Formulate the Equation: (x - 1) + x + (x + 1) = 120 • Solve for x: Notice how the -1 and +1 cancel out effortlessly. 3x = 120 x = 40 • Find Greatest Side: The individual sides are 40-1=39, 40, and 40+1=41. The greatest side is represented by (x + 1) = 41 cm. Exam Strategy & Shortcut: If three consecutive numbers sum up to a specific value, the middle number is always exactly the average of that sum. 120 / 3 = 40. So the consecutive numbers must be 39, 40, and 41. The greatest of these is clearly 41. This allows you to solve the entire problem mentally in just seconds. Common Pitfall: Setting the sides up the traditional way as x, x+1, x+2 (3x+3=120 \Rightarrow 3x=117 \Rightarrow x=39), and then hastily selecting 39 (which is the smallest side) instead of completing the final required step to find the greatest side (x+2 = 41). Final Answer: Therefore, the correct answer is 41 cm.

Question 9

The area of a triangle is $216	ext{ cm}^2$ and its sides are in the ratio $3 : 4 : 5$. The perimeter of the triangle is

Accepted Answer

Concept & Ratio Properties: When the sides of a triangle are in the ratio 3 : 4 : 5, they form a Pythagorean triplet (since 3^2 + 4^2 = 5^2). This means the triangle is a right-angled triangle. The area of a right-angled triangle is given by: Area = (1/2) × Base × Height Step-by-Step Solution: 1. Let the sides of the triangle be 3x, 4x, and 5x. 2. The base and height are the legs of the right triangle: 3x and 4x. 3. Substitute the values into the area formula: 216 = (1/2) × 3x × 4x 4. Simplify the equation: 216 = 6x^2 x^2 = 36 x = 6 5. The sides of the triangle are 3(6) = 18 cm, 4(6) = 24 cm, and 5(6) = 30 cm. 6. Calculate the perimeter: Perimeter = 18 + 24 + 30 = 72 cm. Exam Strategy & Shortcut: Instead of calculating individual sides, note that the perimeter in terms of x is 3x + 4x + 5x = 12x. Once you find x = 6 from the area equation 6x^2 = 216, directly multiply 12 × 6 = 72. This saves calculation time. Common Pitfall: A common mistake is assuming the sides are exactly 3, 4, and 5 and just finding the perimeter of that basic triangle (12 cm). Always use a variable (like x) to represent the common ratio multiplier. Final Answer: Therefore, the correct answer is 72 cm.

Question 10

The sides of a triangle are 5 cm, 6 cm, and 7 cm. One more triangle is formed by joining the mid-points of the sides. The perimeter of the second triangle in cm is

Accepted Answer

Concept & Midpoint Theorem for Perimeter: According to the midpoint theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is exactly half of its length. Consequently, the perimeter of the inner triangle formed by joining the midpoints is exactly half the perimeter of the original triangle. Perimeter_{new} = \frac{Perimeter_{original}}{2} Step-by-Step Solution: 1. Calculate the perimeter of the original triangle by summing its sides: Original Perimeter = 5 + 6 + 7 = 18 cm. 2. The sides of the new triangle are half the lengths of the original sides: New sides = (5/2), (6/2), and (7/2) (which are 2.5 cm, 3 cm, and 3.5 cm). 3. Calculate the perimeter of the new triangle: New Perimeter = 2.5 + 3 + 3.5 = 9 cm. (Alternatively, just divide the original perimeter by 2: (18/2) = 9 cm). Exam Strategy & Shortcut: Save time by finding the sum of the original sides and immediately halving the result. 5 + 6 + 7 = 18 \rightarrow (18/2) = 9. Common Pitfall: A common mistake is trying to divide the area by 4 and then reverse-engineering the side lengths. Perimeter scales linearly (factor of (1/2)), while area scales quadratically (factor of (1/4)). Final Answer: Therefore, the correct answer is 9.

Question 11

A triangle of area $9 y 	ext{ cm}^2$ has been drawn such that its area is equal to the area of an equilateral triangle of side $6	ext{ cm}$. The value of $y$ would be

Accepted Answer

Concept & Equilateral Triangle Area: The area of an equilateral triangle can be determined if the side length is known using the standard formula: Area = \frac{\sqrt{3}}{4} a^2 where a is the side of the triangle. Step-by-Step Solution: 1. Identify the given side length of the equilateral triangle: a = 6 cm. 2. Calculate the area of this equilateral triangle: Area = \frac{\sqrt{3}}{4} × (6)^2 Area = \frac{\sqrt{3}}{4} × 36 = 9\sqrt{3} cm^2. 3. We are given that the area of another triangle is 9 y cm^2 and it equals the area of this equilateral triangle. 4. Equate the two expressions for the area: 9 y = 9\sqrt{3} 5. Solve for y: y = \sqrt{3}. Exam Strategy & Shortcut: Memorize the formula for the area of an equilateral triangle. When setting up the equation 9y = \frac{\sqrt{3}}{4}(36), the 9s immediately cancel out, leaving y = \sqrt{3} instantly. Common Pitfall: A common mistake is confusing the formula for the height of an equilateral triangle (\frac{\sqrt{3}}{2}a) with the formula for the area. Final Answer: Therefore, the correct answer is \sqrt{3}.

Question 12

The hypotenuse of a right-angled isosceles triangle is $5	ext{ cm}$. The area of the triangle is

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Concept & Right-Angled Isosceles Triangle: A right-angled isosceles triangle has two equal perpendicular legs (let's call them a). By the Pythagorean theorem, the hypotenuse h is given by h = a\sqrt{2}. The area of this triangle is given by: Area = (1/2) × a × a = (1/2)a^2 Alternatively, area can be directly related to the hypotenuse: Area = (h^2/4) Step-by-Step Solution: 1. Let the equal legs of the right-angled isosceles triangle be a. 2. The given hypotenuse is 5 cm. 3. Using the relationship h = a\sqrt{2}, we get 5 = a\sqrt{2}, so a = (5/\sqrt{2)}. 4. Substitute a into the area formula: Area = (1/2) × ((5/\sqrt{2)})^2 Area = (1/2) × (25/2) Area = (25/4) = 6.25 cm^2. Exam Strategy & Shortcut: For a right-angled isosceles triangle, memorize the direct shortcut formula relating area and hypotenuse: Area = (h^2/4). Squaring the hypotenuse gives 25, and dividing by 4 gives 6.25 instantly. Common Pitfall: Forgetting to square the \sqrt{2} in the denominator when calculating the area is a common arithmetic error, leading to (25/2) = 12.5. Final Answer: Therefore, the correct answer is 6.25 cm^2.

Question 13

The base and altitude of a right-angled triangle are $12	ext{ cm}$ and $5	ext{ cm}$ respectively. The perpendicular distance of its hypotenuse from the opposite vertex is

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Concept & Properties of Right Triangles: The "perpendicular distance of the hypotenuse from the opposite vertex" is precisely the altitude drawn to the hypotenuse. For a right-angled triangle with perpendicular legs p and b, and hypotenuse h, the length of this altitude is given by: Altitude = (p × b/h) Step-by-Step Solution: 1. Identify the given base (b = 12 cm) and altitude/perpendicular (p = 5 cm). 2. Calculate the hypotenuse (h) using the Pythagorean theorem: h = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 cm. 3. Calculate the perpendicular distance to the hypotenuse: Distance = (12 × 5/13) Distance = (60/13) cm. 4. Convert the improper fraction to a mixed number: (60/13) = 4 with a remainder of 8, which is 4(8/13) cm. Exam Strategy & Shortcut: Recognize 5-12-13 as a standard Pythagorean triplet immediately. The product of the legs divided by the hypotenuse yields (60/13). The nearest multiple of 13 is 13 × 4 = 52, leaving a remainder of 8, pointing straight to 4(8/13). Common Pitfall: A common trap is mistaking the "altitude" given in the problem statement (5 cm) for the altitude to the hypotenuse. Read carefully: it refers to the legs of the right triangle. Final Answer: Therefore, the correct answer is 4(8/13) cm.

Question 14

If the hypotenuse of a right-angled triangle is $41	ext{ cm}$ and the area of the triangle is $180	ext{ sq. cm}$, then the difference between the lengths of the legs of the triangle must be

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Concept & Algebraic Identities in Geometry: We can determine properties of the legs of a right triangle without solving for them individually by utilizing algebraic identities. Let the legs be x and y. From the Pythagorean theorem and area formula: x^2 + y^2 = h^2 xy = 2 × Area We can find the difference (x - y) using the identity: (x - y)^2 = (x^2 + y^2) - 2xy. Step-by-Step Solution: 1. Let the lengths of the legs be x and y. 2. We are given hypotenuse h = 41. Thus, x^2 + y^2 = 41^2 = 1681. 3. We are given Area = 180. Thus, (1/2)xy = 180 \implies xy = 360. 4. We need to find |x - y|. Use the algebraic identity: (x - y)^2 = x^2 + y^2 - 2xy 5. Substitute the known values: (x - y)^2 = 1681 - 2(360) (x - y)^2 = 1681 - 720 = 961 6. Take the square root of both sides: |x - y| = \sqrt{961} = 31 cm. Exam Strategy & Shortcut: Recognizing 41^2 = 1681 quickly is crucial. If you are familiar with standard Pythagorean triplets, you might know 9, 40, 41. Let's verify: (1/2) × 9 × 40 = 180, which matches the given area perfectly. The difference is 40 - 9 = 31. This skip-steps the algebra entirely. Common Pitfall: A common mistake is trying to solve the non-linear system of equations directly via substitution (e.g., y = (360/x)), which creates a complex quadratic equation in x^2 that consumes too much time. Final Answer: Therefore, the correct answer is 31 cm.

Question 15

The perimeter of a right-angled triangle is $60	ext{ cm}$. Its hypotenuse is $26	ext{ cm}$. The area of the triangle is

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Concept & Algebraic Formulation: You can deduce the area of a right triangle from its perimeter and hypotenuse without knowing the individual legs by using the identity: (x + y)^2 = x^2 + y^2 + 2xy Where x and y are the legs. The area is (1/2)xy. Step-by-Step Solution: 1. Let the perpendicular legs be x and y. 2. The perimeter is given as 60 cm, and the hypotenuse is 26 cm. x + y + 26 = 60 \implies x + y = 34. 3. By the Pythagorean theorem: x^2 + y^2 = 26^2 = 676. 4. Square the sum equation from step 2: (x + y)^2 = 34^2 x^2 + y^2 + 2xy = 1156 5. Substitute the sum of squares from step 3: 676 + 2xy = 1156 2xy = 1156 - 676 = 480 xy = 240. 6. Calculate the area: Area = (1/2)xy = (1/2)(240) = 120 cm^2. Exam Strategy & Shortcut: If you observe the hypotenuse is 26, which is 13 × 2, you might recall the 5-12-13 Pythagorean triplet. Scale it by 2 to get 10-24-26. Check the perimeter: 10 + 24 + 26 = 60. It matches! Calculate area directly: (1/2) × 10 × 24 = 120. Common Pitfall: Avoid the trap of substituting y = 34 - x into the Pythagorean theorem, as solving the resulting quadratic equation (x^2 + (34-x)^2 = 676) is tedious and prone to arithmetic errors compared to utilizing algebraic identities. Final Answer: Therefore, the correct answer is 120 cm^2.

Question 16

In an isosceles triangle, the measure of each of the equal sides is 10 cm and the angle between them is 45°. The area of the triangle is

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Concept & Formula: The area of a triangle when two sides and the included angle are given is: Area = (1/2)ab\sin(	heta) Step-by-Step Solution: 1. Here, a = 10 cm, b = 10 cm, and 	heta = 45^\circ. 2. Area = (1/2) × 10 × 10 × \sin(45^\circ). 3. Area = 50 × (1/\sqrt{2)}. 4. Rationalizing the denominator: 50 × \frac{\sqrt{2}}{2} = 25\sqrt{2} cm^2. Exam Strategy & Shortcut: Recognize that (1/\sqrt{2)} is equivalent to \frac{\sqrt{2}}{2}. Mental calculation: 100/2 = 50, 50/\sqrt{2} = 25\sqrt{2}. Common Pitfall: Forgetting the (1/2) in the area formula or substituting the wrong value for \sin(45^\circ). Final Answer: Therefore, the correct answer is 25\sqrt{2} cm^2.

Question 17

If the area of an equilateral triangle is $24\sqrt{3}$ sq. cm, then its perimeter is

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Concept & Formula: For an equilateral triangle of side a: Area A = \frac{\sqrt{3}}{4}a^2 Perimeter P = 3a Step-by-Step Solution: 1. Given Area = 24\sqrt{3}. 2. \frac{\sqrt{3}}{4}a^2 = 24\sqrt{3}. 3. Divide both sides by \sqrt{3}: (a^2/4) = 24. 4. a^2 = 96 \Rightarrow a = \sqrt{96} = \sqrt{16 × 6} = 4\sqrt{6} cm. 5. Perimeter = 3a = 3(4\sqrt{6}) = 12\sqrt{6} cm. Exam Strategy & Shortcut: Write 24\sqrt{3} as \frac{\sqrt{3}}{4} × (96). The side squared is 96, so the side is \sqrt{96}. Multiply by 3 for perimeter: 3\sqrt{96} = 3 × 4\sqrt{6} = 12\sqrt{6}. Common Pitfall: Forgetting to take the square root of 96, or calculating the perimeter as just 4\sqrt{6} (which is the side, not the perimeter). Final Answer: Therefore, the correct answer is 12\sqrt{6} cm.

Question 18

The areas of two equilateral triangles are in the ratio 25 : 36. Their altitudes will be in the ratio

Accepted Answer

Concept & Formula: The area of similar figures is proportional to the square of their corresponding linear dimensions (sides, altitudes, medians). (A_1/A_2) = ((h_1/h_2))^2 Step-by-Step Solution: 1. We are given the ratio of the areas of two equilateral triangles: (A_1/A_2) = (25/36). 2. The ratio of their altitudes h_1 and h_2 is the square root of the ratio of their areas. 3. (h_1/h_2) = \sqrt{(A_1/A_2)} = \sqrt{(25/36)}. 4. (h_1/h_2) = (5/6). Exam Strategy & Shortcut: For any 2D similar shapes (including all equilateral triangles), Area Ratio = (Side Ratio)^2 = (Altitude Ratio)^2. Simply take the square root of 25:36 to get 5:6. Common Pitfall: Assuming altitude ratio is the same as area ratio, or squaring the area ratio instead of taking the square root. Final Answer: Therefore, the correct answer is 5 : 6.

Question 19

An equilateral triangle is described on the diagonal of a square. What is the ratio of the area of the triangle to that of the square?

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Concept & Formula: Diagonal of a square with side s is s\sqrt{2}. Area of an equilateral triangle with side a is \frac{\sqrt{3}}{4}a^2. Step-by-Step Solution: 1. Let the side of the square be s. Area of the square = s^2. 2. The diagonal of this square is s\sqrt{2}. 3. This diagonal is the side of the equilateral triangle. So, triangle side a = s\sqrt{2}. 4. Area of the equilateral triangle = \frac{\sqrt{3}}{4}(s\sqrt{2})^2 = \frac{\sqrt{3}}{4}(2s^2) = \frac{\sqrt{3}}{2}s^2. 5. The ratio of the area of the triangle to that of the square is \frac{\frac{\sqrt{3}}{2}s^2}{s^2} = \frac{\sqrt{3}}{2}. 6. This can be written as \sqrt{3} : 2. Exam Strategy & Shortcut: Assume the side of the square is 1. Its area is 1. Its diagonal is \sqrt{2}. The area of the equilateral triangle on the diagonal is \frac{\sqrt{3}}{4}(\sqrt{2})^2 = \frac{\sqrt{3}}{2}. The ratio is simply \frac{\sqrt{3}}{2} : 1 = \sqrt{3} : 2. Common Pitfall: Squaring s\sqrt{2} incorrectly to get just 2s instead of 2s^2, leading to wrong area values. Final Answer: Therefore, the correct answer is \sqrt{3} : 2.

Question 20

The ratio of bases of two triangles is $x : y$ and that of their areas is $a : b$. Then the ratio of their corresponding altitudes will be

Accepted Answer

Concept & Ratio of Areas of Triangles: The area of a triangle is directly proportional to the product of its base and its altitude (Area = (1/2) × base × altitude). Step-by-Step Solution: • Let the bases of the two triangles be k \cdot x and k \cdot y. • Let their corresponding altitudes be h_1 and h_2. • The ratio of their areas is given as (a/b). • Using the area formula: \frac{Area_1}{Area_2} = (\frac{1/2) \cdot (k \cdot x) \cdot h_1}{(1/2) \cdot (k \cdot y) \cdot h_2} • Substitute the given area ratio: (a/b) = (x \cdot h_1/y \cdot h_2) • Rearranging to solve for the ratio of the altitudes ((h_1/h_2)): (h_1/h_2) = (a \cdot y/b \cdot x) • Therefore, the ratio is ay : bx. Exam Strategy & Shortcut: Altitude is proportional to \frac{Area}{Base}. Therefore, the ratio of altitudes is simply \frac{Ratio of Areas}{Ratio of Bases}. Divide the area ratio terms by the base ratio terms: (a/x) : (b/y), which simplifies to ay : bx by cross-multiplying. Common Pitfall: Accidentally multiplying the ratios directly to get ax : by instead of dividing them. Always set up the standard area fraction first to prevent this. Final Answer: Therefore, the correct answer is ay : bx.

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