Question

m∠B=90 in ΔABC. BM is altitude to AC. If AM=BM=8, find AC

Answer 1

This corollary is used in this question.

If an altitude is drawn to hypotenuse of a right angled triangle ,then the length of altitude is the Geometric mean of length of the segment of hypotenuse formed by the altitude.

BM² = AM × CM

** Geometric mean : geometric mean of two positive numbers a and b is √ab.

SOLUTION :

Given : ∠B = 90°, BM ⟂ AC & AM = BM = 8

In ∆ ABC , ∠B = 90°, BM ⟂ AC

BM² = AM × CM

8² = 8 × CM

64 = 8 CM

CM = 64/8= 8

AC = AM + CM

AC = 8 + 8 = 16

AC = 16

Hence, AC = 16

HOPE THIS ANSWER WILL HELP YOU

Answer 2

In ∆ABC , <B = 90°

BM is altitude of AC .

AM = BM = 8

In ∆ABC and ∆AMB ,

<A = <A ( common angle )

<ABC = <AMB = 90°

∆ABC ~ ∆AMB ( By A.A similarity )--( 1 )

Similarly ∆ABC ~ ∆BMC ------( 2 )

From ( 1 ) and ( 2 ) , we get,

∆AMB ~ ∆BMC

AM/BM = BM/MC

[ If two triangles are similar then

corresponding sides are proportional )

BM² = MC × AM

8² = MC × 8

MC = 8²/8

MC = 8

AC = AM + MC

= 8 + 8

AC = 16

I hope this helps you.

: )