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Heron’s Formula — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsHeron’s Formula4 Marks
Question
A rhombus shaped sheet with perimeter 40cm and one diagonal 12cm, is painted on both sides at the rate of Rs. 5 per m2. Find the cost of painting.
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Answer

Let ABCD be a rhombus having each side equal to x cm.

i.e., AB = BC = CD = DA = x cm

Given, perimeter of a rhombus = 40

$\therefore\text{ AB}+\text{BC}+\text{CD}+\text{DA}=40$

$\Rightarrow\ \text{x}+\text{x}+\text{x}+\text{x}=40$

$\Rightarrow4\text{x}=40$

$\Rightarrow\text{x}=\frac{40}{4}$

$\therefore\ \text{x}=10\text{cm}$

In $\triangle\text{ABC},$ let $\text{a}=\text{AB}=10\text{cm},\text{ b}=\text{BC}=10\text{cm}$

and $\text{c}=\text{AC}=12\text{cm}$

Now, semi-perimeter of a $\triangle\text{ABC},\text{ s}=\frac{\text{a}+\text{b}+\text{c}}{2}$

$=\frac{10+10+12}{2}=\frac{32}{2}=16\text{cm}$

$\therefore\text{ Area of }\triangle\text{ABC}=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ [by Heron's formula]

$=\sqrt{16(16-10)(16-10)(16-12)}$

$=\sqrt{16\times6\times6\times4}$

$=4\times6\times2=48\text{cm}^2$

$\because$ Cost of painting of the sheet of 1cm2 = Rs. 5cm

$\therefore$ Cost of painting of the sheet of 96cm2 = 96 × 5 = Rs. 480

Hence, the cost of the painting of the sheet for both sides = 2 × 480 = Rs. 960

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