# How many right triangles can be constructed?

A right triangle $$PQR$$ is to be constructed in the $$xy$$-plane so that the right angle is at $$P$$ and line $$PR$$ is parallel to the $$x$$-axis. The $$x$$ and $$y$$ coordinates of $$P$$, $$Q$$ and $$R$$ are to be integers that satisfy the inequalities: $$-4\le x\le5$$ and $$6\le y\le16$$. How many different triangles could be constructed with these properties?

a) $$110\quad$$ b) $$1, 100\quad$$ c) $$9, 900\quad$$ d) $$10, 000\quad$$

My try:

$$x=\{-4, -3, -2,\ldots 3, 4,5\}$$ , $$y=\{6, 7, 8,\ldots 14, 15,16\}$$

there are 10 points on the x-axis & 11 points on the y-axis therefore

total number of triangles $$=\binom{10}{2}\binom{11}{1}+\binom{10}{1}\binom{11}{2}=1045$$

But my answer does not match any option. Please correct me if I am wrong. Somebody please help me solve it.
My book suggests that answer must be 9,900

thanks

## 1 Answer

Considering first the number of rectangles should help.

The number of rectangles is given by $$\binom{10}{2}\times\binom{11}{2}=2475$$ For each rectangle, we have four distinct right triangles, so the answer is $$4\times 2475=9900$$