Question

The side lengths of trapezium are $\sqrt[4]{43},\sqrt[4]{43},\sqrt[4]{43}$ and $2\sqrt[4]{43}$. It's area is the ratio of two relatively prime positive integers $m$ and $n$. The value of $(m+n)$ is equal to

• A. $5$
• B. $7$
• C. $9$
• D. $13$

Answer 1

The correct option is D $13$

The area of the trapezium is $=\frac{1}{2}\left(\text{Sum of parallel sides}\right)\times \text{Height}$

$A=\frac{1}{2}(3\times 3^{\frac{1}{4}})\times h$

Now to find $h$, we have to use Pythagorus theorem triangle $\Delta AEF$ where $\angle AEF$ is a right angle.

$h^2+{\left(\sqrt[4]{43}\right)}^2=(\sqrt[4]{43}{)}^2$

$\Rightarrow h^2=\frac{3^{\frac{3}{2}}}{2}$

$\Rightarrow h=\frac{3^{\frac{3}{4}}}{2}$

Area $A=\frac{1}{2}(3\times 3^{\frac{1}{4}})\times h$

$A=\frac{1}{2}\times \left(3^{\frac{5}{4}}\right)\times \left(\frac{3^{\frac{3}{4}}}{2}\right)$

$A=\frac{9}{4}=\frac{m}{n}$

The value of $m$ and $n$ are $9$ and $4$.

The sum of $m$ and $n$ is $13$.