Question

D is the mid-point of side BC of a $\Delta ABC$. AD is bisected at the point E and BE produced cum AC at the point X. Prove that BE EX= 3:1

Answer 1

ABC is a triangle in which D is the midpoint of BC, E is the midpoint of AD. BE produced meets AC at X.

We need to prove BE: EX = 3 : 1

In $△$ BCX and $△$ DCY

$\angle$ CBX = $\angle$ CDY (corresponding angles)

$\angle$ CXB = $\angle$ CYD (corresponding angles)

$△$ BCX∼ $△$ DCY (angle-angle similarity)

We know that corresponding sides of similar sides of similar triangles are proportional

Thus,

(As D is the midpoint of BC)

….(i)

In $△$ AEX and $△$ ADY,

$\angle$AEX = $\angle$ADY (corresponding angles)

$\angle$AXE = $\angle$AYD (corresponding angles)

$△$AEX ∼ $△$ADY (angle-angle similarity)

We know that corresponding sides of similar sides of similar triangles are proportional

Thus,

(As D is the midpoint of BC)

…(ii)

Dividing eqn. (i) by eqn. (ii)

=4
BX = 4EX

BE + EX = 4EX

BE = 3EX

BE : EX = 3:1

Hence Proved