Question

The side lengths of trapezium are $\sqrt[4]{43},\sqrt[4]{43},\sqrt[4]{43}and2\times \sqrt[4]{43}$. Its 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 $13$

$ED$ $=\sqrt[4]{43}$

$DC+FE=\sqrt[4]{43}$

Area of $AEF$ $=\frac{1}{2}xb$

Area of $ABDE$ $=b.\sqrt[4]{43}$

From triangle $AEF$, $x^2+b^2={\left(\sqrt[4]{43}\right)}^2=\sqrt{3}$

$⟹b^2=\sqrt{3}-\frac{\sqrt{3}}{4}=\frac{3\sqrt{3}}{4}⟹b=\frac{\sqrt[4]{43^3}}{2}$

Area of trapezium $=\frac{1}{2}(a+b)h=\frac{1}{2}(2.\sqrt[4]{43}+\sqrt[4]{43})b=\frac{1}{2}\left(3.\sqrt[4]{43}\right)b$

Area of trapezium $=\frac{\sqrt[4]{43^3}}{2}.\frac{\sqrt[4]{43^5}}{2}=\frac{9}{4}$

So, sum of $m$ and $n$ is $13$.