Question

Find the area of a triangle two sides of which are 12 cm and 10 cm and the perimeter is 30cm.​

Answer 1

Answer:$15\sqrt{7} cm^{2}$

Step-by-step explanation:

let the third side be X

perimeter of triangle= side+ side+side

30= 12+10+X

30-12-10=X

X=8cm ##

now ,

         $s=\frac{a +b+c}{2} \\s=\frac{12 +10+8}{2} \\\\s=\frac{30}{2}\\s=15cm$

now using heron's formula

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

$\sqrt{15(15-12)(15-10)(15-8)}\\ =\sqrt{5*3*3*5*7} \\\\=15\sqrt{7}cm^{2}$

Answer 2

Correct Question :

Find the area of a Scalene triangle whose two sides are 12 cm and 19 cm , and the perimeter of the triangle is 30 cm.

Given :

• a = 12 cm
• b = 19 cm
• Perimeter = 30 cm

To find :

The area of the triangle.

Solution :

Since it's a Scalene triangle , we need the three sides of the triangle .

So let's first find the third side of the triangle (c) .

We know the formula for Perimeter of the triangle i.e,

$\boxed{\bf{P = a + b + c}}$

Where ,

• a,b and c are the sides of the triangle

• P = Perimeter of the triangle

Now , using the formula for perimeter of a triangle and substituting the values in it, we get :

$:\implies \bf{30 = 12 + 10 + c}$

$:\implies \bf{30 = 22 + c}$

$:\implies \bf{30 - 22 = c}$

$:\implies \bf{8 = c}$

$\boxed{\therefore \bf{c = 8}}$

Hence the third side (c) of the triangle is 8 cm.

Now to find the area of the triangle :

We know the formula for Area of Scalene triangle i.e,

$\boxed{\bf{A = \sqrt{s(s - a)(s - b)(s - c)}}}$

Where :

• a, b, and c = Sides of the triangle
• s = Semi-perimeter

Here,

Semi perimeter = Perimeter/2

==> s = 30/2

==> s = 15 cm

Hence the Semi-perimeter of the triangle is 15 cm.

Now using the heron's formula and substituting the values in it, we get :

$:\implies \bf{A = \sqrt{15(15 - 12)(15 - 10)(15 - 8)}}$

$:\implies \bf{A = \sqrt{15 \times 3 \times 5 \times 7}}$

$:\implies \bf{A = \sqrt{15 \times 3 \times 5 \times 7}}$

$:\implies \bf{A = \sqrt{1575}}$

$:\implies \bf{A = 15\sqrt{7}}$

$\boxed{\therefore \bf{A = 15\sqrt{7}\:cm^{2}}}$

Hence, the area of the triangle is 15√7 cm²