Question

if you draw two right triangles using the line as the hypotenuse, do the triangles have to be similar? Why or why not?

Answer 1

Answer:

All the right triangles will be similar.

Step-by-step explanation:

See the attached diagram.

Let us assume any two right triangles are drawn on the line OP, and AB and A'B' as the hypotenuse on the same line OP.

Since the slope of the line OP is constant and assume that it makes x° with the positive x-axis.

Hence, $\tan x = \frac{AC}{BC} = \frac{A'C'}{B'C'}$

⇒ $\frac{AC}{A'C'} = \frac{BC}{B'C'}$

Therefore, the sides of the two right triangles are in the same ratio and hence, Δ ABC and Δ A'B'C' are similar.

Hence, all the right triangles will be similar. (Answer)

Answer 2

Answer:

Yes - they have to be similar.  Since they are using the same line as the hypotenuse, the ratio of the other two sides are in the same ratio (ie, the slope of the line), the 3 inner angles will be the same.  Thus, the triangles will be similar.

Step-by-step explanation: