Question

In △RST, point Q is on side RS . Prove that if ST>RT then TQ<ST

Answer 1

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,

`ST^2=TH^2+SH^2.`

In triangle QTH, by the Pythagorean theorem,

`QT^2=TH^2+QH^2.`

Subtract these two equations:

`ST^2-QT^2=SH^2-QH^2.`

In this case SH>QH, then
`SH^2-QH^2>0,` thus
`ST^2-QT^2>0` and, consequently,
`ST^2>QT^2\Rightarrow ST>QT.`

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,

`ST^2=TH^2+SH^2.`

In triangle QTH, by the Pythagorean theorem,

`QT^2=TH^2+QH^2.`

Subtract these two equations:

`ST^2-QT^2=SH^2-QH^2.`

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

• triangle is isosceles, then ST=RT and the height TH is also the median. This means that SH=RH;
• ST>RT, then SH>RH
• ST<RT, then SH<RH.

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

Therefore,
`SH^2-QH^2>0,` thus
`ST^2-QT^2>0` and, consequently,
`ST^2>QT^2\Rightarrow ST>QT.`