Question

In triangle ABC, if `AD_|_BC` and `AD^2 = BD*DC`. Then find the angle BAC

Answer 1

GIVEN `A Delta ABC` in which `AD bot BC and AD^(2)=BD*CD`.
PROOVE `angle BAC=90^(@)`

PROOF `AD^(2)=BD*CD rArr (BD)/(AD)=(AD)/(CD)`
Now, in `Delta DBA and Delta DAC`, we have
`angle BDA= angle ADC=90^(@) and (BD)/(AD)=(AD)/(CD)`
`:. Delta DBA ~ Delta DAC` [ by SAS- similarity]
`:. angle B= angle 2 and angle1 = angleC`
`:. angle 1+angle 2= angleB+ angle C rArr angle A= angle B+ angleC`
`rArr 2angle A = angleA + angle B+ angleC =180^(@)`
`rArr angleA= angleBAC= 90^(@)`