PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR . Show that PM^2 = QM . MR.

Step 1: Use the given data to make a diagram.
GIVEN: ΔPQR is right angled at P is a point on QR such that PM ⊥QR.
Step 2: Prove that the triangles (PRM and PQM) are similar
NOTE: \angle PMQ = \angle PMR = 90
\angle a + \angle b = 90.............(1)
\angle a + \angle c = 180 - 90 = 90 ................(2)
From equation (1) and (2) we can write
\angle b = \angle c
Similarly, \angle a = \angle d
Therefore, by using the AA similarity
[math] \triangles PRM , \triangle PQM are similar
Step 3: Find the required ratio by using the similar triangle properties.
EXAMPLE: ΔPQM similar ΔPMR
Therefore,
The corresponding sides are proportional
\frac{QM}{PM} = \frac{PM}{RM}
Cross multiply
QM.RM = PM. PM
PM^2 = QM. RM