Question

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

a. 9√15 sq.cm

b. 6√15 sq.cm

c. 3√15 sq.cm

d. √15, sq.cm.

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Answer 1

Given :-

• Two equal sides of an isosceles triangle = 12 cm
• Perimeter = 30 cm

To find :-

• Area of triangle = ?

Solution :-

Let one side of an isosceles triangle be x.

As we know,

• Perimeter = Sum of all the sides.

• So, By adding the given sides and subtracting from the perimeter. We get the measure of the third side.

$\mapsto$ x + 12 + 12 = 30

$\mapsto$ x + 24 = 30

$\mapsto$ x = 30 - 24

$\mapsto$ x = 6 cm

• Third side = 6 cm

• Now we have the measure of all the sides. By using Heron's formula. We will get the area of the triangle.

• $\huge\sf\boxed{s~=~\dfrac{a + b + c}{2}}$

$\mapsto$ $\sf{s~=~\dfrac{12+ 12 + 6}{2}}$

$\mapsto$ $\sf{s~=~\dfrac{30}{2}}$

• $\sf{s~=~15~cm}$

$\huge\sf \boxed{Heron's \: formula = \sqrt{s(s - a)(s - b)(s - c)}}$

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

$\sf { = \sqrt{15(3)(3)(9)}}$

$\sf { = \sqrt{3 \times 5 \times 3 \times 3 \times 3 \times 3}}$

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

• Area = 9$\sf{\sqrt{15}cm^2}$

Hence, Option A is correct.

Answer 2

Let the other side be X cm

Perimeter of triangle = sum of all sides

30 = 12 +12 + X

X = 6 cm

S= semiperimeter = perimeter/2

S= 15

a=12 cm

b= 12 cm

c= 6 cm

area =√{s(s-a)(s-b)(s-c)}

= √{15(15-12)(15-12)(15-6)}

= √{15*3*3*9}

= √{9²*15}

= 9√15 cm²