In △rst, point q is on side rs . prove that if st> rt then tq< st

Consider triangle RST. In this triangle ST>RT and point Q is on the side RS.

Draw the height TH to the side RS. There are two cases where point H can lie.

1 case: Point H lies to the right from points S and R (see first diagram).

Now consider two right triangles STH and QTH.

In triangle STH, by the Pythagorean theorem,

In triangle QTH, by the Pythagorean theorem,

Subtract these two equations:

In this case SH>QH, then thus and, consequently,

2 case: Point H lies between points S and R (see second diagram).

Consider two right triangles STH and QTH.

In triangle STH, by the Pythagorean theorem,

In triangle QTH, by the Pythagorean theorem,

Subtract these two equations:

In this case, the height falls into point H that divides side SR into two parts SH and RH. Note that if

You have that ST>RT, then SH>RH and SH>QH.  

Therefore, thus and, consequently,