Question

The perimeter of an isosceles triangle is 32 cm. The ratio of equal side to it's base is 3:2.find the area of triangke

Answer 1

$\bold\green{\underline{Correct\: Question:}}$

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.

$\huge\bold\purple{\underline{Solution:}}$

$\bold\green{\underline{Given:}}$

The perimeter of isosceles Triangle=32cm

The ratio of the equal sides of its base is 3:2

$\large{\underline{\orange{\rm{Let:}}}}$

The ratio of equal sides and base be X

The length of equal sides be 3x

The length of base be 2x

$\large{\underline{\red{\rm{A.T.Q:}}}}$

$\sf{\dashrightarrow Perimeter\:of\: Triangle=32}$

$\sf{\dashrightarrow 3x+3x+2x=32}$

$\sf{\dashrightarrow 8x=32}$

$\sf{\dashrightarrow 8x=32}$

$\sf{\dashrightarrow x=4}$

• Now sides of Triangle are

$\sf{\dashrightarrow equal\: sides=3x=3*4=12\:cm}$

$\sf{\dashrightarrow base=2x=2*4=8\:cm}$

• Find semi perimeter here
• a=12 b=12 c=8

$\sf{\dashrightarrow semi\: perimeter=\frac{a+b+c}{2}}$

$\sf{\dashrightarrow semi\: perimeter=\frac{12+12+8}{2}}$

$\sf{\dashrightarrow semi\: perimeter=\frac{32}{2}}$

$\sf{\dashrightarrow semi\: perimeter=16}$

• Using heron's formula here

$\large{\green{\red{\rm{Area\:of\:Triangle=\sqrt{s(s-a)(s-b)(s-c)}}}}}$

$\sf{\dashrightarrow Area=\sqrt{16(16-12)(16-12)(16-8)}}$

$\sf{\dashrightarrow Area=\sqrt{16*4*4*8}}$

$\sf{\dashrightarrow Area=32\sqrt{2}\:cm^2}$

Hence,

The area of isosceles Triangle=32√2cm²

Answer 2

${\purple{\underline{\underline{\huge{\mathtt{Answer:}}}}}}$

Given:

We have been given that the perimeter of an isosceless triangle is 32cm. And also the ratio of its equal side to its base is

3 : 2 .

To Find:

We need to find the area of triangle.

Solution:

As it is given that the ratio if its equal side to its base is 3 : 2, which means that the length of equal side is 3x and that of base is 2x.

We know that the perimeter of an isosceles triangle is 2 × length of equal side + base.

=> 2 × 3x + 2x = 32cm

=> 6x + 2x = 32cm

=> 8x = 32cm

=> 32/8 = x

=> 4cm = x

or x = 4cm

Now, length of equal side = 3x = 3 × 4 = 12cm.

We know that the area of an isosceles triangle is

$b \times \frac{1}{4} \times \sqrt { {4a}^{2} - {b}^{2} }$

Substituting the values we have,

$= 8 \times \frac{1}{4} \times \sqrt{4 \times 144 - 64}$

$= \sqrt[2]{512}$

$= 2 \times 22.62$

$= 45.25 {cm}^{2}$

Therefore the area of triangle is 45.25cm^2.