Question

In ΔABC, AL and CM are the perpendiculars from the vertices A and C to BC and AB respectively. If AL and CM intersect at O, prove that :
(i) ΔOMA~ ΔOLC
(ii) OA/OC = OM/OL

Answer 1

Hence it is proved that,

(i) ΔOMA~ ΔOLC

(ii) OA/OC = OM/OL

Consider the figure while going through the following steps:

Given,

In ΔABC, AL and CM are the ⊥s from the vertices A and C to BC and AB.

AL and CM intersect at O

⇒ AL ⊥BC

CM ⊥ AB

In Δ OMA and Δ OLC

∠ AMO = ∠ CLO (angles are equal to 90°)

∠ MOA = ∠ LOC (vertically opposite angles)

By AA similarity, we have,

∴ Δ OMA ~ Δ OLC

As corresponding sides of similar triangles are proportional, we have,

∴ OA/OC = OM/OL