Ex.12.1 Q3 Heron’s Formula Solution - NCERT Maths Class 9


Question

There is a slide in a park. One of its side walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” see Figure.  If the sides of the wall are \(15 \)m, \(11 \)m and \(6 \)m, find the area painted in colour.

Text Solution

What is known?

Sides of the wall i.e. Dimensions of the triangle.

What is unknown?

Area of the (triangle) i.e. area of slope painted in colour.

Reasoning:

By using Heron’s formula we can calculate the area of triangle.

The formula given by Heron about the area of a triangle

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

Where \(a, b\) and \(c\) are the sides of the triangle, and

\[\begin{align}s &= \text{Semi-perimeter}\\& = \begin{Bmatrix} \text{Half the Perimeter } \\ \text{ of the triangle} \;\end{Bmatrix}  \\&=\frac{(a+b+c)}{2}\end{align}\]

Steps:

The sides of the walls (triangle) are \(a =11\; {\rm{m}} , b = 6 \, {\rm{m}}\;\text{and} \; c=15\, {\rm{m}}. \) 

Semi Perimeter:

\(\begin{align}s & =\frac{(a+b+c)}{2} \\ & =\frac{11+6+15}{2} \\ &=\frac{32}{2} \\ &=16 \mathrm{m}\end{align}\)

By using Heron’s formula,

Area of a triangle  

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

Area of a triangle wall:

\(\begin{align}&=\sqrt{s(s-a)(s-b)(s-c)} \\ &=\!\sqrt{\!16(16\!-\!11)(16\!-\!6) (16\!-\!15)  } \\ &=\sqrt{16 \times 5 \times 10 \times 1} \\ &=\sqrt{800} \mathrm{m}^{2} \\ &=20 \sqrt{2} \mathrm{m}^{2}\end{align}\)

Area of the wall of park painted in color \( = 20 \sqrt{2} \;\rm{m^2}\)

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