Question

∠B is a right angle in ΔABC. BD 1 AC and D ∈ AC. If AD = 4DC, prove that BD = 2DC.

Answer 1

According to Geometric mean theorem.

In right angle triangle if an altitude is perpendicular to hypotenuse

and divides into p and q parts .Then the length of altitude h can be given as

$h = \sqrt{pq}$

Now, proceed as shown below.

Answer 2

In ∆ ABC , <B = 90°

and BD is perpendicular to AC

AD = 4DC

RTP : AD = 4DC

proof :

In ∆ABC , and ∆ ADB ,

<A = <A ( common angle )

<ABC = <ADB = 90°

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

Similarly ∆ABC ~ ∆BDC -------( 2 )

From ( 1 ) and ( 2 ) we get ,

∆ADB ~ ∆BDC

Therefore ,

DA/BD = BD/DC

[ If two triangles are similar then

corresponding sides are proportional ]

BD² = DC × DA

BD² = DC × 4( DC ) [ given ]

BD² = 4× DC²

BD² = ( 2 × DC )²

BD = 2DC

Hence proved .

I hope this helps you.

: )