D is a point on side QR of ΔPQR

Question:

D is a point on side QR of ΔPQR such that PD ⊥ QR. Will it be correct to say that ΔPQD ~ ΔRPD? Why?

Solution:

False

In ΔPQD and ΔRPD,

PD = PD                                                 [common side]

∠PDQ = ∠PDR                                                    [each 90°]

Here, no other sides or angles are equal, so we can say that ∠PQD is not similar to ΔRPD. But, if ∠P = 90°,

then ∠DPQ = ∠PRD

[each equal to 90° – ∠0 and by ASA similarity criterion, ΔPQD ~ΔRPD]

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