Question 1

The height of a triangle is equal to the perimeter of a square whose diagonal is $8\sqrt{2}$ metre and the base of the same triangle is equal to the side of a square whose area is 729sq. metre. What is the area of the triangle? (In sq. metre)  [Unit…

Accepted Answer

Concept & Multi-Shape Integration: This problem connects the properties of squares (diagonal, perimeter, area, side) to the dimensions of a triangle (base, height) to calculate the triangle's area. Diagonal_{square} = a\sqrt{2} Area_{square} = s^2 Area_{triangle} = (1/2) × base × height Step-by-Step Solution: • Step 1: Find the triangle's height. • The diagonal of the first square is 8\sqrt{2} m. • Formula for diagonal is side × \sqrt{2}. Thus, the side of the first square is 8 m. • The perimeter of this square is 4 × 8 = 32 m. • Given: Height of triangle = Perimeter of this square = 32 m. • Step 2: Find the triangle's base. • The area of the second square is 729 sq. m. • Formula for area is side^2 = 729. Taking the square root, the side is \sqrt{729} = 27 m. • Given: Base of triangle = Side of this square = 27 m. • Step 3: Calculate the area of the triangle. • Area = (1/2) × base × height. • Area = (1/2) × 27 × 32. • Area = 27 × 16. • 27 × 16 = 27 × (10 + 6) = 270 + 162 = 432 sq. m. Exam Strategy & Shortcut: Since you need to multiply (1/2) × 27 × 32, you can just calculate the unit digit. The unit digit of 27 × 16 is 7 × 6 = 42, which ends in 2. Looking at the options, both (b) 206 and (c) 472 and (d) 432 end in 2, but 27 × 16 is close to 25 × 16 = 400, making 432 the only logical choice. Common Pitfall: Confusing the area of the square with its side or perimeter, or forgetting the (1/2) multiplier in the triangle area formula. Final Answer: Therefore, the correct answer is 432.

Question 2

A courtyard is 25m long and 16m broad is to be paved with bricks of dimensions 20cm by 10cm. What is the total number of bricks required?

Accepted Answer

Concept & Unit Conversion and Area: The number of bricks required to pave an area is calculated by dividing the total area to be paved by the surface area of a single brick. Crucially, all dimensions must be in the same units (either all meters or all centimeters). Number\ of\ Bricks = (Total\ Area/Area\ of\ One\ Brick) Step-by-Step Solution: • Given: Courtyard dimensions are 25 m by 16 m. Brick dimensions are 20 cm by 10 cm. • Convert courtyard dimensions to centimeters (1 m = 100 cm). • Length = 25 × 100 = 2500 cm. • Breadth = 16 × 100 = 1600 cm. • Calculate the total area of the courtyard: Area = 2500 × 1600 = 4,000,000 cm^2. • Calculate the area of a single brick: Area = 20 × 10 = 200 cm^2. • Calculate total bricks: Number of bricks = (4,000,000/200). • Divide to find the result: (40,000/2) = 20000. • A total of 20000 bricks are required. Exam Strategy & Shortcut: You can set up the calculation as a single fraction and cancel zeroes before multiplying to avoid dealing with massive numbers: (2500 × 1600/20 × 10) = (250 × 160/2 × 1) = 250 × 80 = 20000. Common Pitfall: Failing to convert the meters into centimeters before calculating the area. If you simply calculate (25 × 16/20 × 10), you will get 2, which is completely incorrect by a magnitude of 10,000 (since 1 m^2 = 10,000 cm^2). Final Answer: Therefore, the correct answer is 20000.

Question 3

The base of an isosceles is 14 cm and its perimeter is 36 cm. Find its area.

Accepted Answer

Concept & Formula: The perimeter of a triangle is the sum of all its sides. For an isosceles triangle with equal sides a and base b, the perimeter is 2a + b. Heron's formula can be used to find the area if all three sides are known: Area = \sqrt{s(s-a)(s-b)(s-c)} where s is the semi-perimeter. Alternatively, a specific formula for isosceles triangles is: Area = (b/4)\sqrt{4a^2 - b^2} Step-by-Step Solution: 1. Given: Base (b) = 14 cm, Perimeter = 36 cm. Let the two equal sides be a. 2. Find Equal Sides: 2a + 14 = 36 2a = 22 \Rightarrow a = 11 cm. The sides are 11, 11, and 14. 3. Calculate Semi-perimeter (s): s = (36/2) = 18 cm. 4. Calculate Area using Heron's Formula: Area = \sqrt{18 × (18-11) × (18-11) × (18-14)} Area = \sqrt{18 × 7 × 7 × 4} Area = \sqrt{9 × 2 × 49 × 4} Area = 3 × 7 × 2 × \sqrt{2} = 42\sqrt{2} sq. cm. Exam Strategy & Shortcut: Instead of Heron's formula, using the altitude is faster if you know Pythagorean triplets, but here (b/2) = 7 and hypotenuse is 11. Altitude h = \sqrt{11^2 - 7^2} = \sqrt{121 - 49} = \sqrt{72} = 6\sqrt{2}. Area = (1/2) × base × height = (1/2) × 14 × 6\sqrt{2} = 42\sqrt{2}. This requires less calculation. Common Pitfall: Misidentifying which side is the base versus the equal sides when applying the perimeter formula leads to the wrong lengths for the unknown sides. Final Answer: Therefore, the correct answer is 42\sqrt{2} sq. cm.

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