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A triangle and a parallelogram have the same base and the same area. If the sides of triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram.
Given:
A triangle and a parallelogram have the same base and the same area.
The sides of triangle are $26\ cm, 28\ cm$ and $30\ cm$, and the parallelogram stands on the base $28\ cm$.
To do:
We have to find the height of the parallelogram.
Solution:
The sides of the triangle are $a=26\ cm, b=28\ cm$ and $c=30\ cm$.
The semi-perimeter of the triangle $s=\frac{a+b+c}{2}$
$=\frac{28+26+30}{2}$
$=42\ cm$
Therefore, by Heron's formula,
$A=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{42(42-28)(42-26)(42-30)}$
$=\sqrt{42(14)(16)(12)}$
$=\sqrt{112896}$
$=336\ cm^2$
Area of the parallelogram is equal to the area of the triangle.
Area of parallelogram $=$ Area of triangle
Base $\times$ corresponding height $=336\ cm^2$
$\Rightarrow 28 \times$ corresponding height $=336\ cm^2$
$\Rightarrow\ Height =\frac{336}{28}$
$=12\ cm$
Hence, the height of the parallelogram is $12\ cm$.
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