Right triangle trigonometry help?

Question

I've got a right triangle where I know the slope of side $c$ based on the two points $(-150,200)$ and $(0,0)$. Also I know the length of side $a$. I was wondering based on these two known factors how would I find the length of side $d$?

Answer 1

That's a rather strange way to draw a triangle. But a diagonal line cuts a rectangle into two congruent triangles, so presumably the $d$ you indicated is the side of one of those triangles although the other side is not drawn.

Remember what slope is: it is rise over run, where rise and run can be found by ... examining a right triangle whose legs are parallel to the axes. So $a$ is the rise and $d$ is the run, except that since the line is rising to the left and falling to the right you have negative slope and it's probably easier to say that $-a$ is the rise and $d$ is the run.

So, $\dfrac{-a}{d} = \text{slope}.$ Plug in the known values of $a$ and the slope and solve for $d$.

Another approach using the same principles but more geometric: draw a right triangle with hypotenuse from $(0,0)$ to $(-150,200)$ and with legs parallel to the legs $a$ and $d$. This new triangle is similar to the triangle with hypotenuse $c$, but all the sides are scaled up by the same factor. Find that factor and find the dimensions of the new triangle, then determine what value of $d$ scales up to the corresponding leg of the new triangle.

Answer 2

From points A (-150, 200) and B (0,0) you have the two sides of the big triangle measuring 200 (height) and 150 (width) respectively. The smaller triangle is just a scaled version of the big triangle, so: d/a = (width/height), given you the value d = a * (width/height).