Question

In ΔABC, m∠A=90, AD is an altitude. So AB² = .....(a) BD.BC
(b) BD.DC
(c) BD/DC(d) BC.DC

Answer 1

∆ABC is a right - angled triangle in which angle A is 90° . D is point lies on BC in such a way that AD is altitude on BC.

from right angled ∆ABD ,
AB is hypotenuse ,
from Pythagoras theorem,
AB² = AD² + BD² -----(1)

similarly, from right angled ∆ACD ,
AC is hypotenuse ,
from Pythagoras theorem,
AC² = AD² + CD² ------(2)

from equations (1) and (2),
2AD² = AB² - BD² + AC² - CD²
= AB² + AC² - (BD² + CD²) -----(3)

we know,from ABC right angled triangle,
BC² = AB² + BC². put it equation (3),

2AD² = BC² -(BD² + CD²)
= (BD + CD)² - (BD² + CD²)
= BD² + CD² + 2BD.CD - (BD² + CD²)
= 2BD.CD
hence, AD² = BD.CD
therefore, option (b) is correct.