In a △ ABC, D is the midpoint of side AC such that BD = ½ AC. Show

1.	In a △ ABC, D is the midpoint of side AC such that BD = ½ AC. Show that ∠ ABC is a right angle.
Answer» From the figure we know that D is the midpoint of the line AC So we get AD = CD = ½ AC It is given that BD = ½ AC So we can write it as AD = BD = CD Let us consider AD = BD We know that the angles opposite to equal sides are equal So we get ∠ BAD = ∠ ABD …..(1) Let us consider CD = BD We know that the angles opposite to equal sides are equal So we get ∠ BCD = ∠ CBD ….. (2) By considering the angle sum property in △ ABC We get ∠ ABC + ∠ BAC + ∠ BCA = 180^o So we can write it as ∠ ABC + ∠ BAD + ∠ BCD = 180^o By using equation (1) and (2) we get ∠ ABC + ∠ ABD + ∠ CBD = 180^o So we get ∠ ABC + ∠ ABC = 180^o By addition 2 ∠ABC = 180^o By division ∠ABC = 90^o Therefore, ∠ABC is a right angle.

In a △ ABC, D is the midpoint of side AC such that BD = ½ AC. Show that ∠ ABC is a right angle.

Answer»

From the figure we know that D is the midpoint of the line AC

So we get

AD = CD = ½ AC

It is given that BD = ½ AC

So we can write it as

AD = BD = CD

Let us consider AD = BD

We know that the angles opposite to equal sides are equal

So we get

∠ BAD = ∠ ABD …..(1)

Let us consider CD = BD

We know that the angles opposite to equal sides are equal

So we get

∠ BCD = ∠ CBD ….. (2)

By considering the angle sum property in △ ABC

We get

∠ ABC + ∠ BAC + ∠ BCA = 180^o

So we can write it as

∠ ABC + ∠ BAD + ∠ BCD = 180^o

By using equation (1) and (2) we get

∠ ABC + ∠ ABD + ∠ CBD = 180^o

So we get

∠ ABC + ∠ ABC = 180^o

By addition

2 ∠ABC = 180^o

By division

∠ABC = 90^o

Therefore, ∠ABC is a right angle.