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Heron’s Formula — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsHeron’s Formula1 Mark
Question
Write True or False and justify your answer:In a triangle, the sides are given as 1$1\ cm, 12\ cm$ and $13\ cm$. The length of the altitude is $10.25\ cm$ corresponding to the side having length $12\ cm.$
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Answer

We have the length of the altitude corresponsing to the side having length $12\ cm$
$2\text{s}=11\text{cm}+12\text{cm}+13\text{cm}=36\text{cm}$
$\Rightarrow\text{s}=36\div2=18\text{cm}$
$\text{Area of }\triangle=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$
$=\sqrt{18(18-11)(18-12)(18-13)}$
$=\sqrt{18\times7\times6\times5}$
$=\sqrt{2\times3\times3\times7\times2\times3\times5}$
$=2\times3\sqrt{105}$
$=6\sqrt{105}\text{cm}^2$
$\text{Length of altitude}=\frac{2\text{Area of }\triangle}{\text{Base}}$
$=\frac{2\times6\sqrt{105}}{12}=\sqrt{105}=10.25\text{cm}$
Hence, the given statement is true.

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