Question

1. Find the cost of levelling the ground in the form of a triangle having the sides 51 m, 37 m and 20 m at the rate of ₹ 3 per m². 2. Find the area of the isosceles triangle whose perimeter is 11 cm and the base is 5 cm. 3. Find the area of the equilateral triangle whose each side is 8 cm. 4. The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3 : 2. Find the area of the triangle.

Answer 1

1. Find the cost of levelling the ground in the form of a triangle having the sides 51 m, 37 m and 20 m at the rate of ₹ 3 per m².

a = 51 m, b = 37 m, c = 20 m

semiperimeter: p = (51+37+20):2 = 54 m

Area of triangle:

`A=√(p(p-a)(p-b)(p-c))\\\\A=√(54(54-51)(54-37)(54-20))\\\\A=√(54\cdot3\cdot17\cdot34)\\\\A=√(9\cdot2\cdot3\cdot3\cdot17\cdot17\cdot2)\\\\A=3\cdot2\cdot3\cdot17\\\\A=306\,m^2`

Rate: ₹ 3 per m².

Cost: ₹ 3•306 = ₹ 918

2. Find the area of the isosceles triangle whose perimeter is 11 cm and the base is 5 cm.

a = 5 cm

a+2b = 11 cm ⇒ 2b = 6 cm ⇒ b = 3 cm

p = 11:2 = 5.5

`A=√(5.5(5.5-3)^2(5.5-5))\\\\ A=√(5.5\cdot(2.5)^2\cdot0.5)\\\\ A=√(11\cdot0.5\cdot(2.5)^2\cdot0.5)\\\\A=0.5\cdot2.5\cdot√(11)\\\\A=1.25√(11)\,cm^2\approx4.146\,cm^2`

3. Find the area of the equilateral triangle whose each side is 8 cm.

a = b = c = 8 cm

p = (8•3):2 = 12 cm

`A=√(12(12-8)^3)\\\\ A=√(12\cdot4^3)\\\\ A=√(3\cdot4\cdot4\cdot4^2)\\\\A=4\cdot4\cdot√(3)\\\\A=16\sqrt3\ cm^2\approx27.713\ cm^2`

4. The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3 : 2. Find the area of the triangle.

a = 2x

b = 3x

2x + 2•3x = 32 cm ⇒ 8x = 32 cm ⇒ x = 4 cm ⇒ a = 8 cm, b = 12 cm

p = 32:2 = 16 cm

`A=√(16(16-8)(16-12)^2)\\\\ A=√(16\cdot8\cdot4^2)\\\\ A=√(2\cdot8\cdot8\cdot4^2)\\\\ A=8\cdot4\cdot\sqrt2\\\\ A=32\sqrt2\ cm^2\approx45.2548\ cm^2`