Question

the base of an isosceles triangle is 12cm and its perimeter is 32 cm.Find its area​

Answer 1

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Given :-

• Base = 12 cm
• The triangle has two equal sides (isosceles)
• Perimeter is 32 cm

To find :-

• Area ?

SOLUTION :-

The perimeter of a triangle = a + b + c

Two sides are equal, So, perimeter = 2a + b

Base given = 12 cm

So, 2a + 12 = 32

a = 10 cm.

The formula to find the area of a triangle is :-

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

Finding s ...

$= \dfrac{10 + 10 + 12}{2}$

$= 32 \div 2$

$= 16 \: cm$

So, $Area= \sqrt {16(16-10)(16-10)(16-12)}$

$Area= \sqrt {16\times 6\times 6\times 4}$

$Area= \sqrt {2\times 2 \times 2 \times 2 \times 3 \times 2 \times 3 \times 2 \times 2\times 2}$

$Area= 24 {cm}^{2}$

__________

Answer 2

$\bf{\underline{\underline{Answer:}}}$

$\bf Area\: of \:isosceles\: \triangle = 24 {cm}^{2}$

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

• Base of isosceles traingle = 12 cm
• Its Perimeter = 32 cm

$\bold{\underline {To\:Find:}}$

• Area of isosceles triangle = ?

$\bf{\underline{\underline{Step\: by\: step \:explanation:}}}$

In Isosceles triangle two sides are equal.

So , other two sides be x = x = 2x

Perimeter = 32 cm

Value of x ⟹ sum of all side = perimeter

$\therefore \:\:\:a + b + c = 32\\ \\</p><p>\implies x + x + 12 = 32\\ \\</p><p>\implies 2x + 12 = 32\\ \\</p><p>\implies 2x = 32 - 12\\ \\</p><p>\implies 2x = 20\\ \\</p><p>\implies x = \dfrac{20}{2} \\ \\</p><p>\implies x = 10$

Now, the formula to find the semi-perimeter of a triangle is :-

$\bigstar {\boxed {\bf S = \dfrac{ a+ b+ c}{2}}}$

Substituting value in above formula, we get

$S = \dfrac{ 10 + 10+ 12}{2}\\ \\</p><p>=\dfrac{32}{2}\\ \\= 32 \div 2\\ \\</p><p></p><p>= 16 \: cm$

Now, by Heron's formula find the area of isosceles triangle :

$</p><p></p><p>\bigstar{\boxed{\bf Area \:of \triangle = {\sqrt{s(s-a) (s-b) (s-c)}}}}$

Substituting value in above formula, we get,

$Area= \sqrt {16(16-10)(16-10)(16-12)}</p><p> \\ \\</p><p> </p><p></p><p>Area= \sqrt {16\times 6\times 6\times 4}</p><p> \\ \\</p><p> </p><p></p><p>Area= \sqrt {2\times 2 \times 2 \times 2 \times 3 \times 2 \times 3 \times 2 \times 2\times 2}\\ \\</p><p></p><p>Area= {2\times 2 \times 2 \times 2 \times 3}</p><p></p><p>\\ \\</p><p> </p><p> \bf Area= 24 {cm}^{2}$

Thus, area of isosceles triangle = 24 cm²