Question

The equal aides of the
isosceles triangle are
12 cm, and the perimeter is 30cm. The
area of this triangle is:
​

Answer 1

Given :-

• The equal side of the isosceles triangle is 12 cm.

• Perimeter of the isosceles triangle is 30 cm

Have to find :-

• What's the area of the triangle.

Solution :-

Let's find out the third side of the isosceles triangle first.

So, we consider the unknown side of the triangle → x cm.

As we know the formula of perimeter of triangle.

Applying it here.

• P = a + b + c [ sum of all it's sides ]

Now, substituting the values in the above formula,

So,

p → " 30 "

a → " 12 "

b → " 12 "

c → " x "

$:\implies \bf{P = a + b + c} \\ \\ \\ :\implies \bf{30 = 12 + 12 + x} \\ \\ \\ :\implies \bf{30 - 24 = x} \\ \\ \\ :\implies \bf{6 = x} \\ \\ \\ \boxed{\therefore \bf{x = 6}}$

Then, The third side is " x " → 6 cm

____________________________

Now , Applying the formula for area of a isosceles triangle i.e,

$\boxed{\bf{A = \dfrac{1}{4}b\sqrt{4a^{2} - b^{2}}}}$

Here, we know that

b = Base / Third side of triangle.

a = Equal side of triangle.

Substituting the values in the above formula :

$:\implies \bf{A = \dfrac{1}{4}b\sqrt{4a^{2} - b^{2}}}$

$:\implies \bf{A = \dfrac{1}{4} \times 6 \times \sqrt{(4 \times 12^{2}) - 6^{2}}}$

$:\implies \bf{A = \dfrac{1}{\not{4}^{2}} \times \not{6}^{3} \times \sqrt{(4 \times 12 \times 12) - 6^{2}}}$

$:\implies \bf{A = \dfrac{3}{2} \times \sqrt{(4 \times 144) - 36}}$

$:\implies \bf{A = \dfrac{3}{2} \times \sqrt{576 - 36}}$

$:\implies \bf{A = \dfrac{3}{2} \times \sqrt{576 - 36}}$

$:\implies \bf{A = \dfrac{3}{2} \times \sqrt{540}}$

$:\implies \bf{A = \dfrac{3}{2} \times \sqrt{2 \times 2 \times 3 \times 3 \times 15}}$

$:\implies \bf{A = \dfrac{3}{2} \times 2 \times 3 \sqrt{15}}$

$:\implies \bf{A = \dfrac{3}{2} \times 6\sqrt{15}}$

$:\implies \bf{A = \dfrac{3}{\not{2}} \times \not{6\sqrt{15}}}$

$:\implies \bf{A = 3 \times 3\sqrt{15}}$

$:\implies \bf{A = 9\sqrt{15}}$

$\underline{\boxed{\therefore \bf{A = 9\sqrt{15}\:cm^{2}}}}$

Thus ,

Our Final answer { the area of triangle } → 9√15 cm².

Answer 2

Answer:

Correct Question:–

1) An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

To find,

• Area of the triangle

Given as,

Perimeter of isosceles triangle = 39cm

Each side of the triangle = 12cm

( We have to use Herons formula to find the area of the triangle )

Solution:-

⟹ a + b + c = 30

⟹ 24 + c = 30

⟹ c = 6cm

⟹$S=\frac{30}{2} =15 cm$

Herons formula:–

$area = \sqrt{s(s - a)(s-b)(s-c} \\ \\ = \sqrt{15(15-12)(15-12)(15-6)} \\ = \sqrt{15×3²×9} = 9 \sqrt{15} {cm}^{2}$

Hope it helps uh!