Question

ABC is a triangle in which `AB = AC and D` is a point on AC such that `BC^2 = AC xx CD`.Prove that `BD = BC`.

Answer 1

GIVEN ` A Delta ABC` in which `AB=AC and D` is a point on AC such that `BC^(2) =ACxxDC`.
TO PROOVE BC=BC
PROOF `BC^(2)=ACxxDC` (given)
`rArr (BC)/(DC)=(AC)/(BC)`
Thus, in `Delta ABC and DeltaBDC` , We have
`(BC)/(DC)=(AC)/(BC)= and angle C= angle C` (common)
`:. Delta ABC~ Delta BDC` [ by SAS- Siimilarity]
`rArr (AC)/(BC)=(AB)/(BD)`
`rArr (AC)/(BC)=(AC)/(BD) [ :. AB=AC ("given")]`
`rArr BD= BC`.
Hence, BD=BC.