Question

. The perimeter of an isosceles triangle is 42 cm and its base is 3/2each of the equal sides. Find (i) the length of each side of the
(ii) the area of the triangle, and (iii) the height of the triangle.​

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\red{Answer:}}}}}}$

Given:-

• The perimeter of an isosceles triangle = 42cm
• Base of the triangle = 3/2 times each of equal sides.

To find :-

• the length of each side of
• the area of the triangle
• the height of the triangle.

Solution :-

Let's assume the two sides of the isosceles triangle as x

The third side as per the given condition in the question = $\bf\large\frac{3x}{2}$

Perimeter = 42 cm

To find the length of side x we will use the formula of perimeter of a triangle.

Perimeter of a triangle = x + x + $\bf\large\frac{3x}{2}$

42 = 2x + $\bf\large\frac{3x}{2}$

42 = $\bf\large\frac{2 (2x) + 3x}{2}$

42 = $\bf\large\frac{ 4x + 3x}{2}$

42 = $\bf\large\frac{7x}{2}$

7x = 42 × 2

7x = 84

x = $\large\frac{84}{7}$

x = 12 cm

Length of two congruent sides of the isosceles triangle = 12 cm

Length of third side = $\bf\large\frac{3(12)}{2}$

Length = $\bf\large\frac{36}{2}$

Length = 18 cm

•°• Three sides of the isosceles triangle are

1. 12 cm = a
2. 12 cm = b
3. 18 cm = c

I assumed them to be, a, b and c just for the convenience in calculation.

Now, the first thing we need to calculate is the area of the isosceles triangle. For this we will use the Heron's formula.

Since, we are provided with the perimeter of the isosceles triangle in the question, we will directly divide it by 2 to find the value of s.

s = $\bf\large\frac{42}{2}$

s = 21 cm

Area of triangle using Heron's formula,

$\bf\sqrt{s(s - a)(s - b)(s - c)}$

$\bf\sqrt{21 ( 21-12)(21-12)(21-18)}$

$\bf\sqrt{21(9)(9)(3)}$

$\bf\sqrt{21(81) (3)}$

$\bf\sqrt{21 (243)}$

$\bf\sqrt{5103}$

Area = 27√7 cm²

Now, the last thing. We need to find the height of the isosceles triangle.

For this, as we have the area of the triangle (27√7 cm²)and base (18cm) of the triangle, we can use the formula : $\bf\large\frac{1}{2}$ × base × height

Area of triangle = $\bf\large\frac{1}{2}$ × base × height

27√7 = $\bf\large\frac{1}{2}$ × 18 × height

27√7 = 9 × height

$\bf\large\frac{27√7}{9}$ = height

3√7 = height

•°• length of sides of the triangle are

• 12 cm, 12cm, 18cm

Area of the triangle is,

• 27√7 cm²

Height of the triangle is,

• 3√7 cm

$\bf{\large{\underline{\boxed{\mathfrak{\purple{Verification:}}}}}}$

For sides of the triangle :-

Given, Perimeter = 42 cm

Perimeter is the sum of all sides, so if we get the sum of all three sides (12,12,18) equivalent to 42, then we can assure that the our answer is right.

Perimeter = x + x + $\bf\large\frac{3x}{2}$

42 = 12 + 12 + 18

42 = 24 + 18

42 = 42

LHS = RHS

Hence our sides are correct.

For the height of the triangle :-

Area of the triangle = 27√7 cm²

Area of triangle = $\bf\large\frac{1}{2}$ × base × height

Plug in the values,

27√7 = $\large\frac{1}{2}$ × 18 × 3√7

27√7 = 9 × 3√7

27√7 = 27√7

LHS = RHS

Hence, the height of the triangle too is correct. Similarly it even gives us the assurity that our area of the triangle too is correct.