Question

PQR is a triangle right angled at P and M is a point on QR such that PM perpendicular QR show that PM²=QM×MR​

Answer 1

Answer:

Given:-

• In triangle PQR angle p = 90°
• PM perpendicular to QR

RTP:-

• PM²=QM × MR

Proof:-

In triangle RPQ and RMP

${ \sf{{ \angle{r} = { \angle{r}} \: \: (common \: angle)}}}$

${ \sf{{ \angle{RPQ}} = { \angle{RMP}} = 90°}}$

Triangle RPQ ~ RMP (by AA similarity)..... (1)

Similarly, Triangle RPQ ~ PMQ..... (2)

From(1) & (2) we get:-

Triangle RMP ~ PMQ

${ \sf{ \therefore{ \frac{MR}{PM} = \frac{PM}{QM} }}}$

(if two triangles are similar then corresponding sides are propotional)

${ \therefore{ \sf{PM² = QM × MR}}}$