SOLUTION: Given- Isosceles triangle ABC with segment Ab congruent to segment AC line CD is perpendicular to AB. Prove- Segment BC multiplied by segment BC is equal to two times segment AB

Algebra ->  Geometry-proofs -> SOLUTION: Given- Isosceles triangle ABC with segment Ab congruent to segment AC line CD is perpendicular to AB. Prove- Segment BC multiplied by segment BC is equal to two times segment AB       Log On


   

Question 251610: Given- Isosceles triangle ABC with segment Ab congruent to segment AC
line CD is perpendicular to AB.
Prove- Segment BC multiplied by segment BC is equal to two times segment AB multiplied by segment BD
Answer by vksarvepalli(154) About Me  (Show Source):
You can put this solution on YOUR website!
consider the triangle BCD which is a right angled triangle
now using Pythagoras theorem we get
BC%5E2=CD%5E2+%2B+BD%5E2------------1
now consider the triangle ACD which is also a right angled triangle
again using Pythagoras th.
AC%5E2=AD%5E2%2BCD%5E2------------2
but AB=AC (isosceles)
so substituting it in 2 we get
AB%5E2=AD%5E2%2BCD%5E2
but AB=AD+DB
so, %28AD%2BBD%29%5E2=AD%5E2%2BCD%5E2
i.e., BD%5E2%2BAD%5E2%2B2%2AAD%2ABD=AD%5E2%2BCD%5E2
on simplifying we get CD%5E2=BD%5E2%2B2%2AAD%2ABD
substituting the value of CD^2 in equation 1
BC%5E2=BD%5E2%2B2%2AAD%2ABD+%2B+BD%5E2
taking out BD common in the right hand side of the equation
BC%5E2=2%2ABD%2A%28AD%2BBD%29
but AD+BD=AB
therefore BC%5E2=2AB%2ABD