Q: A triangle with three equal sides has its area equal to 3√3 sq cm . Find its perimeter .

According to the question, √3a2/4 = 3√3 [ side = a, and area = √3a2/4] ⇒ a2/4 = 3 ⇒ a2 = 3 x 4 ∴ a = 2√3 ∴ Required perimeter = 3a =3 x 2√3 = 6√3 cm

Question 1

The area of a trapezium is 384 sq.cm. If its parallel sides in ratio 3 : 5 and the perpendicular distance between them be 12 cm. The smaller of parallel sides is?

Accepted Answer

Let the parallel sides be 3a and 5a. So Area of trapezium = 1/2 x sum of parallel side x perpendicular distance between them. ⇒ 1/2 (3a +5a) x 12 = 384 ⇒ 8a = 64 ∴ a =8 ∴ Smaller side = 3x = 3 x 8 = 24 cm.

Question 2

The two parallel sides of a trapezium are 1 meters and 2 meters respectively . The perpendicular distance between them is 6 meters. The area of the trapezium is?

Accepted Answer

Area of trapezium = 1/2 x sum of parallel sides x perpendicular distance between them = 1/2 (1 + 2) x 6 m2 = 9 m2

Question 3

A parallelogram has sides 60 m and 40 m and one of the diagonal is 80 m long. Then its area is?

Accepted Answer

AB = 60 m, BC = 40 m and AC = 80 m ∴ s = (60 + 40 + 80 ) / 2 m = 90 m (s-a) = 90 - 60 = 30 m, (s-b) = 90 - 40 = 50 m and (s-c) = 90 - 80 = 10 m ∴ Area of Δ ABC = √s(s-a)(s-b)(s-c) = √90 x 30 x 50 x 10 m2 = 300√15 m2 ∴ Area of parallelogram ABCD = 2 x area of Δ ABC = 600√15 m2

Question 4

The adjacent sides of parallelogram are 6 cm and 4 cm and the angle between then is 30°. The area of the parallelogram is?

Accepted Answer

AB = 6 cm ; AD = 4 cm and ∠ BAD = 30° Area of parallelogram ABCD = AB x AD x sin 30° = 6 x 4 x sin 30° = 12 cm2

Question 5

One side of a parallelogram is 14 cm. Its distance from the opposite side is 16 cm. The area of the parallelogram is?

Accepted Answer

Area of parallelogram = Side of parallelogram x distance from the opposite side = 14 x 16 cm2 = 224 cm2

Question 6

The length of the diagonal of a rhombus is 80% of the length of the other diagonal, Then the area of the rhombus is how many times the square of the length of the longer diagonal?

Accepted Answer

Let length of the longer diagonal = d cm Then, length of other diagonal = 0.8 x d cm Area of rhombus = (1/2) x d x 0.8 x d = 2/5 d2 = 2/5 d2 Area of square of the length of the longer diagonal = d2 So the area of the rhombus is 2/5 times the square of the length of the longer diagonal.

Question 7

In a rhombus, whose area is 144 sq.cmone of its diagonal is twice as long as the other, The lengths of its diagonals are?

Accepted Answer

Area of rhombus = (d1 x d2) / 2 ⇒ (d x 2d ) / 2 = 144 ⇒ d2 = 144 ⇒ d = 12 ∴ Length of diagonal = 12 cm, 24 cm

Question 8

The lengths of three line segments (in cm) are given in each of the four cases. Which one of the following cases is not suitable to be the three sides of a triangle?

Accepted Answer

We know that in any triangle "the sum of two sides is always greater than its third side" and "the difference of two sides is always less than its third side". (i) 2 + 3 is not greater than 5 (ii) |5 - 2| not less than 3

Question 9

The three sides of a triangle are 15, 25 and x units. Which one of the following is correct?

Accepted Answer

In a triangle Sum of two sides is always greater than 3rd side i.e., x < 25 + 15 = 40 .....(i) Difference of two sides is always less than 3rd side i.e., 25 - 15 = 10 < x ...(ii) From Eqs. (i) and (ii) , we get 10 < x < 40

Question 10

A triangle with three equal sides has its area equal to 3√3 sq cm . Find its perimeter .

Accepted Answer

According to the question, √3a2/4 = 3√3 [ side = a, and area = √3a2/4] ⇒ a2/4 = 3 ⇒ a2 = 3 x 4 ∴ a = 2√3 ∴ Required perimeter = 3a =3 x 2√3 = 6√3 cm

Question 11

The sides of a triangle area in the ratio of 1/3 : 1/4 : 1/5 and its perimeter is 94 cm. Find the length of the smallest side of the triangle .

Accepted Answer

Given ratio = 1/3 : 1/4 : 1/5 = 20 : 15 : 12 Let length of the sides be 20k, 15k and 12k. Then, according to the question, 20k + 15k + 12k = 94 ⇒ 47k = 94 ∴ k = 94/47 = 2 Smallest side = 12k = 12 x 2 = 24 cm

Question 12

The perimeter of an equilateral triangle is 90 cm. Find its area.

Accepted Answer

Given that, perimeter = 90 cm ⇒ 3a = 90 cm [a = side] ∴ a = 90/3 = 30 cm ∴ Area = √3a2 / 4 = √3/4 x (30)2 = 900 x √3/4 = 225√3 sq cm

Question 13

The area of an equilateral triangle is √ 243 /4 sq cm. Find the length of its side.

Accepted Answer

According to the question, area = √3a2/4 = √243 /4 ⇒ a2 = √81 x 3/√3 ∴ a = √9 = 3 cm

Question 14

The diagonals of a squares is 4√2 cm. The diagonal of another square whose area is double that of the first square is

Accepted Answer

Diagonal of square = √2a [a = side] 4√2 = √2 a a = 4 cm Now, area of square = a2 = (42) = 16 Side of a square whose area is 2 x 16. a12 = 32 ⇒ a1 = √32 ⇒a14√2 Now, diagonal of new square = √2a = √2x 4 √2 = 8 cm

Question 15

The diagonals of two squares are in the ratio of 3 : 2. Find the ratio of their areas.

Accepted Answer

Let the diagonals of the squares be 3x and 2x. ∴ Ratio of their areas = [(1/2) (3x2)] / [(1/2) (2x2)] = 9/4

Question 16

If the sides of a squares is increased by 25%, then the area of the squares will be increased by

Accepted Answer

Required increment = 2a + [a2 / 100] % = 2 x 25 + [(252)/100)] % = 50 + (625/100)% = 56.25%

Question 17

The perimeter of two square is 12 cm and 24 cm. The area of the bigger square is how many times that of the smaller?

Accepted Answer

Ratio of area of 2 squares = (ratio of perimeter of 2 squares)2 = (24/12)2 = 4

Question 18

A cost of cultivating a square field at a rate of ₹ 135 per hectare is ₹ 1215. The cost of putting a fence around it at the rate of 75 paise per meter wouldbe

Accepted Answer

Area of the field =1215/135 = 9 hec = 90000 m2 [1 hec =10000 m2] ∴ Side of the field = √90000 = 300 m Perimeter of the field = 4 x 300 = 1200 m Now, cost of putting a fence around field = (1200 x 75)/100 = ₹ 900

Question 19

Diagonals of a rhombus area 1 m and 1.5 m in lengths. The area of the rhombus is

Accepted Answer

Area of rhombus =(d1 x d2)/2 =(1 x 1.5)/2 m2 = 0.75 m2

Question 20

If the diagonals of a rhombus are 4.8 cm 1.4 cm, then what is the perimeter of the rhombus?

Accepted Answer

Perimeter of rhombus = 2 √ d1 2 + d2 2 = 2 √(4.8)2 + (1.4)2 = 2 √23.04 + 1.96 = 2 √25 = 2 x 5 = 10 cm

Area Questions