Exam

Olympiad workbook

Subject

IOQM

Chapter

Straight Lines

Olympiad workbook · IOQM · Straight Lines

Consider a square A B C D of side length 16 . Let E, F be points on...

Consider a square $A B C D$ of side length 16 . Let $E, F$ be points on $C D$ such that $C E=E F=F D$. Let the line BF and $A E$ meet in $M$. The area of $\triangle M A B$ is:

Step-by-step solution

Eq of line $A E: 2 y-3 x+16=0$
Eq of line $B F$ : $2 y+3 x-32=0$ $\Rightarrow M \equiv(8,4)$ Area of triangle $A M B$ $$\begin{aligned}\left|\frac{1}{2}\right| \begin{array}{ccc}0 & 16 & 1 \\ 16 & 16 & 1 \\ 8 & 4 & 1\end{array}|\mid & =\frac{1}{2}(-16(8)+1(64-8 \times 16)) \\ & =\left|\frac{1}{2}(-16 \times 16+64)\right| \\ & =\left|\frac{16}{2}(4-16)\right|=8 \times 12 \\ & =96 \text { sq.units }\end{aligned}$$