Question
The parallel sides of a trapezium are $20\ cm$ and $10\ cm.$ Its nonparallel sides are both equal, each being $13\ cm.$ Find the area of the trapezium.
✓
Answer
In trapezium $ABCD,$
$AB || DC$ and $AD = BC $
$AB = 20\ cm, CD = 10\ cm.$
$AD = BC = 13\ cm,$
Through $C$, draw $CE || DA$ and $\text{CF}\perp\text{EB}$ or $AB,$
Then $CE = CA = 13\ cm.$
$ EB = AB - AE = AB - CD = 20 - 10 = 10\ cm.$
Now side of $\triangle\text{ECB}$ are $13, 13, 10\ cm,$
$\text{s}=\frac{13+13+10}{2}=\frac{36}{2}=18$
$\therefore$ area of $\triangle\text{ECB},$
$=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ (Hero's formula)
$=\sqrt{18(18-13)(18-13)(18-10)}$
$=\sqrt{18\times5\times5\times8}=\sqrt{3600}=60\text{cm}^2$
But area of $\triangle\text{CEB}=\frac{1}{2}\times\text{EB}\times\text{CF}$
$\Rightarrow60=\frac{1}{2}\times10\times\text{CF}$
$\Rightarrow\text{CF}=\frac{60\times2}{10}=12\text{cm}.$
$\therefore$ Distance between two parallel lines $(h) = 12\ cm.$
$\therefore$ Area of trapezium $=\frac{1}{2}\text{h}(\text{l}_1+\text{l}_2)$
$=\frac{1}{2}\times12(20+10)\text{cm}^2$
$=6\times30=180\text{cm}^2$
$AB || DC$ and $AD = BC $
$AB = 20\ cm, CD = 10\ cm.$
$AD = BC = 13\ cm,$
Through $C$, draw $CE || DA$ and $\text{CF}\perp\text{EB}$ or $AB,$
Then $CE = CA = 13\ cm.$
$ EB = AB - AE = AB - CD = 20 - 10 = 10\ cm.$
Now side of $\triangle\text{ECB}$ are $13, 13, 10\ cm,$
$\text{s}=\frac{13+13+10}{2}=\frac{36}{2}=18$
$\therefore$ area of $\triangle\text{ECB},$
$=\sqrt{\text{s}(\text{s}-\text{a})(\text{s}-\text{b})(\text{s}-\text{c})}$ (Hero's formula)
$=\sqrt{18(18-13)(18-13)(18-10)}$
$=\sqrt{18\times5\times5\times8}=\sqrt{3600}=60\text{cm}^2$
But area of $\triangle\text{CEB}=\frac{1}{2}\times\text{EB}\times\text{CF}$
$\Rightarrow60=\frac{1}{2}\times10\times\text{CF}$
$\Rightarrow\text{CF}=\frac{60\times2}{10}=12\text{cm}.$
$\therefore$ Distance between two parallel lines $(h) = 12\ cm.$
$\therefore$ Area of trapezium $=\frac{1}{2}\text{h}(\text{l}_1+\text{l}_2)$
$=\frac{1}{2}\times12(20+10)\text{cm}^2$
$=6\times30=180\text{cm}^2$
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