Question

the perimeter of isosceles triangle is 42 cm. its base is 2/3 times the sum of the equal sides.
find the length of each side and area of the triangle

Answer 1

Let, the length of each equal side of the isosceles triangle be x cm.

So, its base will be = $\dfrac{2}{3}$ of ( x + x )

Base = $\dfrac{2}{3}$ × ( 2x )

Base = $\dfrac{4x}{3}$

Perimeter = 42 cm ( Given )

Now,

Using formula,

Perimeter = Sum of all sides

•°•

42 = $\dfrac{4x}{3}$ + x + x

42 = $\dfrac{4x + 3x + 3x}{3}$

42 = $\dfrac{10x}{3}$

x = $\dfrac{42 × 3}{10}$

x = $\dfrac{126}{10}$

x = 12.6

So, length of each equal side = 12.6 cm

Base = $\dfrac{4x}{3}$

Base = $\dfrac{4 × 12.6}{3}$

Base = 16.8 cm

Now,

To find the area, we draw a perpendicular from the meeting point of two equal sides on the base.

So, it divides the base into equal parts, i.e. Each half part = $\dfrac{16.8}{2}$ = 8.4 cm

Applying Pythagoras Theorem,

(Length of equal side)² = (Length of half part of base)² + (Length of the perpendicular)²

Putting values, we get

(12.6)² = (8.4)² + (Perpendicular)²

Perpendicular = $\sqrt{(12.6)^2 - (8.4)^2}$

Perpendicular = $\sqrt{158.76 - 70.56}$

Perpendicular = $\sqrt{88.2}$

Perpendicular = 9.39 cm = 9.4 cm (rounded)

Area of the isosceles triangle =>

= $\dfrac{1}{2}$ × Half part of base × Perpendicular

= $\dfrac{1}{2}$ × 8.4 × 9.4

= $\dfrac{8.4 × 8.4}{2}$

= 35.28 cm²