Question

E is a point on the side BC produced of an issoceles triangle ABC with AB=AC if AD is perpendicular to BC AND EF PERPENDICULAR to AC prove that AB * EF = AD * EC​

Answer 1

Refer to the above attachment.

Answer 2

Given:--- ABC is an isosceles triangle, AD is perpendicular to BC ... BC is produced to E and EF is perpendicular to AC..

To Prove :--- AB * EF = AD * EC

Proof : ----

[ Refer to image First ..]

Given that ABC is an isosceles triangle where AB = AC

So,

→ ∠ABD = ∠ECF [ Angle opposite to Equal sides Are Equal ]

Now, In ΔABD and ΔECF,

∠ADB = ∠EFC (Each 90°)

∠ABD = ∠ECF (Proved above)

Therefore,

ΔABD ~ ΔECF (By using AA similarity criterion)

[ AA Criterion: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. ]

Now,

By CPCT we get,

→ AC/AD = EC/EF

Cross - Multiply we get,

→ AC × EF = AD × EC

Now, since it is given that ∆ABC is a isosceles ∆ with AB = AC ,

So, Replacing AC with AB , we get,

→ AB × EF = AD × EC (Hence, Proved).