Question

the perimeter of a triangle is 30 cm and the circumference of a circle is 88 CM if the area if the area of triangle is 30k cm square then find k

Answer 1

Question:

The perimeter of a triangle is 30 cm and the circumference of the incircle is 88 cm. If the area of the triangle is 30k cm square, then find k.

Solution:

*See attachment

Let the triangle be ΔABC

Let the centre of the circle be O

Find the radius:

Circumference of a circle = 2πr

Given that the circumference is 88 cm

88 cm = 2πr

r = 88 ÷ 2π

r = 14 cm

Find the area of the triangle:

ΔABC =  ΔAOB +  ΔBOC +  ΔCOA

Note that the height of each of the three triangles = radius

Area of triangle = 1/2 x base x height

ΔABC =  (1/2 x AB x 14) +  (1/2 x BC x 14) + (1/2 x AC x 14)

ΔABC = 7 AB + 7 BC + 7 AC

ΔABC = 7 (AB + BC + AC)

Given that the perimeter is 30 cm

Substitute AB + BC + AC = 30:

ΔABC = 7 (30) cm²

Find the value of k:

Area of triangle = 30k (Given)

Area of triangle = 30 (7) cm²

⇒ k = 7

Answer: k = 7