Question

AD and BE are perpendiculars of Isosceles triangle ABC if a C equals BC then prove that a equals BD.
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Answer 1

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Q. AD and BE are perpendiculars of Isosceles triangle ABC if AC=BC then prove that AE=BD.

Ans. Given AD and BE are altitude and AC = BC

Here

∠ BEA = ∠ BEC = 90° -----1

And

∠ ADB = ∠ ADC = 90° -----2

So from equation 1 and 2 , we can say

∠ BEA = ∠ ADB = 90° -----3

And As given ABC is a isosceles triangles so , from base angle theorem ,we can say that

∠ CAB = ∠ CBA ------4

Now In ∆ BAE and ∆ ABD

∠ BEA = ∠ ADB ( From equation 3 )

∠ EAB=∠ DBA(As ∠ CAB =∠ EAB ( same angles )

∠ CBA = ∠ DBA (same angles) And from equation 4 we know ∠CAB = ∠ CBA

AB = AB ( Common side )

Hence

∆ BAE ≅∆ ABD ( By AAS rule )

So,

AE=BD ( By CPCT rule )

Hence proved

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