Question

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

Answer 1

Given :-

Two sides of the triangle (which are equal) = 12 cm

Perimeter of the triangle = 30 cm

To Find :-

The area of the triangle.

Analysis :-

Find the third side of the triangle by adding the given side and subtracting it from the perimeter.

Then find the semi perimeter accordingly.

Using Heron's formula, substitute the values we got and find the area accordingly.

Solution :-

We know that,

• p = Perimeter
• a = Area
• s = Semi perimeter

Making an equation,

30 - (12 + 12)

= 30 - 24

= 6 cm

Therefore, the third side of the triangle is 6 cm.

Finding the semi perimeter,

$\underline{\boxed{\sf Semi \ perimeter=\dfrac{Perimeter}{2} }}$

Given that,

Perimeter (p) = 30 cm

Substituting their values,

Semi perimeter = 30/2

= 15 cm

Therefore, the semi perimeter is 15 cm.

Using Heron's formula,

$\underline{\boxed{\sf Area \ of \ triangle = \sqrt{s(s-a)(s-b)(s-c)} }}$

Substituting their values,

$\sf =\sqrt{15(15-12)(15-12)(15-6)}$

$\sf =\sqrt{15 \times 3 \times 3 \times 9}$

$\sf =9\sqrt{15} \ cm^2$

Therefore, the area of the triangle is 9√15 cm².

Answer 2

$\Large{\boxed{\underline{\overline{\mathfrak{\star \: Question :- \: \star}}}}}$

❥ The equal sides of the isosceles triangle are 12cm, and the perimeter is 30cm.The area of the triangle is :

$\Large{\underline{\underline{\bf{AnSweR:-}}}}$

$✒ \underline{ \boxed{ \sf Area \: of \: \triangle = 9 \sqrt{15} \: cm ^{2} }}$

$\Large{\underline{\underline{\bf{GiVen:-}}}}$

☞ 2 equal sides of the triangle = 12 cm

☞ Perimeter = 30 cm

$\Large{\underline{\underline{\bf{To Find :-}}}}$

☞ The area of isosceles triangle.

$\Large{\underline{\underline{\bf{SoLuTion:-}}}}$

✒ We know that ,

$\large{\boxed{\bold{perimeter\: of\:triangle\:=\:a+b+c}}}$

✒ a ↠ 12 cm.

✒ b ↠ 12 cm.

✒ c ↠ ?

$<strong>✎</strong> \sf \:30 = 12 + 12 + c$

$<strong>✎</strong> \sf \:c = 30 - 24$

$<strong>✦</strong> \underline{ \boxed{ \sf \:c = 6 \: cm}}$

❥ Therefore,

✒ The third side of isosceles triangle is 6 cm.

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❥ Now,

$<strong>✎</strong> \sf Semi - Perimeter \: of \: \triangle = \frac{a + b + c}{2}$

$<strong>✎</strong> \sf Semi - Perimeter \: of \: \triangle = \frac{12 + 12 + 6}{2}$

$<strong>✎</strong> \sf Semi - Perimeter \: of \: \triangle = \frac{30}{2}$

$\underline{ \boxed{ <strong>✎</strong> \sf Semi -Perimeter \: of \: \triangle = 15 \: cm}}$

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$<strong>✎</strong> \sf Area \: of \: \triangle = \sqrt{s(s - a)(s - b)(s - c)}$

✒s ( semi - perimeter ) ↠15 cm

✒a ↠12 cm

✒b ↠12 cm

✒c ↠ 6 cm

Now ,

put the values in above given formula

$<strong>✎</strong> \sf Area \: of \: \triangle = \sqrt{15(15 - 12)(15 - 12)(15 - 6)}$

$<strong>✎</strong> \sf Area \: of \: \triangle = \sqrt{15 \times 3 \times 3 \times 9}$

$<strong>✎</strong> \sf Area \: of \: \triangle = 3 \sqrt{135} \: cm ^{2}$

$✒ \underline { \boxed{ \sf Area \: of \: \triangle = 9 \sqrt{15} \: cm ^{2} }}$

Hence,

❃ The area of isosceles triangle is 9√15 cm².

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