Exam

Olympiad workbook

Subject

IOQM

Chapter

Straight Lines

Olympiad workbook · IOQM · Straight Lines

Let A B C D be a rectangle in which A B+B C+ C D=20 and A E=9 where...

Let $A B C D$ be a rectangle in which $A B+B C+ C D=20$ and $A E=9$ where $E$ is the mid-point of the side $B C$. Find the area of the rectangle.

Step-by-step solution

$$\begin{aligned} & 2 x+y=20 \\ & y=20-2 x ...(i)\\ & \text { In } \triangle A B E \end{aligned}$$ $x^2+\left(\frac{y}{2}\right)^2=9^2$...(ii) Using (i) in (ii) $$\begin{aligned} & x^2+\left(\frac{20-2 x}{2}\right)^2=81 \\ & x^2+(10-x)^2=81 \\ & 2 x^2-20 x+19=0 \\ & x=\frac{20 \pm \sqrt{400-4 \times 2 \times 19}}{2 \times 2} \\ & x=\frac{10 \pm \sqrt{62}}{2} \end{aligned}$$ From (i) $$\begin{aligned} \Rightarrow \quad y & =20-2\left(\frac{10 \pm \sqrt{62}}{2}\right) \\ y & =10 \mp \sqrt{62} \end{aligned}$$ $$\begin{aligned} \text { Area of rectangle }=x y & =\frac{1}{2}(10 \pm \sqrt{62})(10 \mp \sqrt{62}) \\ & =\frac{100-62}{2} \\ & =19\end{aligned}$$