Question

Calculate the area of the shaded region​

Answer 1

To calculate:

The area of the shaded region

Given :

The Shaded region is a shape that is made by cutting out a smaller triangle from the bigger triangle.

hence, the area of the shaded region :

= Area of bigger tri. - Area of smaller tri.

• The smaller triangle is a right angle triangle
• the base = 12 cm
• height = 5 cm
• hypotenuse = ???

• The bigger triangle is a scalene triangle
• First side = 15 cm
• second side = 14 cm
• third side (hypotenuse of previous triangle) = ???

Formula to be applied

$\bold{ \dag \: area = \frac{1}{2} \times b \times h} \\ \bold{ \dag \: area = \sqrt{s(s - a)(s - b)(s - c)}}$

where :

• b= base of right angle triangle
• h = height of right angled triangle
• a = side of triangle
• b = second side of triangle
• c = third side of triangle
• s = semi perimeter

Solution:

1] The area of smaller triangle:

The Triangle is a right angle triangle,

so, Area = 1/2 × base × hight

$\implies \bold{area = \frac{1}{ \cancel2} \times \cancel{12} \times 5}$

$\implies \bold{area = 6 \times 5}$

$\implies \bold{area = 30 \: {cm}^{2} }$

hence, the area of triangle = 30 cm ²

The hypotenuse of the triangle is unknown , so we have to find it to find the value of third side of bigger triangle:

Pythagorean Theorem :

• hypotenuse² = base²+height²

$\implies \bold{hypotenus {e}^{2} = {12}^{2} + {5}^{2} }$

$\implies \bold{hypotenus {e}^{2} = 144 + 25}$

$\implies \bold{hypote {nuse}^{2} = 169}$

$\implies \bold{hypotenuse = 13cm}$

hence, the hypotenuse = 13 cm

2] Area of bigger triangle:

The triangle is a scalene triangle , with sides

• First side = 15 cm
• second side = 14 cm
• third side = 13 cm

semi perimeter = (15+14+13)/2

= 42/2

= 21 cm

so, the area :

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

$\implies \bold{area = \sqrt{21(21 - 15)(21 - 14)(21 - 13)}}$

$\implies \bold{area = \sqrt{21 \times 6 \times 7 \times 8}}$

$\implies \bold{area = \sqrt{3 \times 7 \times 3 \times 2 \times 7 \times 2 \times 2 \times 2}}$

$\implies \bold{area = 2 \times 2 \times 3 \times 7}$

$\implies \bold{area = 84 \: c {m}^{2} }$

so, area of the bigger triangle = 84 cm²

hence, area of the shaded region

= 84 cm² - 30 cm²

= 54 cm²

hence, the area of the shaded region = 54 cm²