Question

in triangle abc angle ACB is equal to 90 degree seg cd perpendicular side ab and seg ce is angle bisector of angle ACB then prove that AD upon BD is equal to AE square upon BE square

Answer 1

Solution:-

Angle bisector theorem:

When vertical angle of a triangle is bisected, the bisector divides the base into two segments which have the ratio as the order of other two sides.

From diagram, it is clear that

ΔADC and ΔCDB are similar.

Then,

AD/ CD = CD/BD = AC/BC -----(I)

since CE is the bisector of ∠ACB,

by angle bisector theorem,

BE/ AE = BC/AC

taking reciprocal

AE/BE = AC/BC

Both side square

AE^2/ BE^2 = AC^2/ BC^2

AE^2/ BE^2 = AC/ BC× AC/ BC

AE^2/ BE^2 = AD/ CD × CD/ BD

AE^2 \ BE^2 = AD / BD

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@GauravSaxena01