In a triangle ABC, D is the mid-point of side AC such that BD = 1/2 AC. Show that ∠ABC is a right angle.

Solution:

Given, ABC is a triangle.

D is the midpoint of side AC

BD = 1/2 AC ------------------------ (1)

We have to show that ∠ABC is a right angle.

D is the midpoint of AC.

AD = CD

AC = AD + CD

Now, AC = AD + AD or CD + CD

AC = 2AD or 2CD

So, AD = CD = 1/2 AC ----------- (2)

Comparing (1) and (2),

AD = CD = BD --------------------- (3)

Considering triangle DAB,

From (3), AD = BD

We know that the angles opposite to the equal sides are equal.

∠ABD = ∠BAD ---------------------------- (4)

Considering triangle DBC,

From (3), BD = CD

We know that the angles opposite to the equal sides are equal.

∠BCD = ∠CBD ---------------------------- (5)

Considering triangle ABC,

By angle sum property,

∠ABC + ∠BAC + ∠ACB = 180°

From the figure, ∠BAC = ∠BAD

∠ACB = ∠DCB

Now, ∠ABC + ∠BAD + ∠BCD = 180°

From (4) and (5),

∠ABC + ∠ABD + ∠CBD = 180°

Given, BD = 1/2 AC

∠ABC = ∠ABD + ∠CBD

Now, ∠ABC + ∠ABC = 180°

2∠ABC = 180°

∠ABC = 180°/2

∠ABC = 90°

Therefore, it is proven that ∠ABC = 90°

✦ Try This: In right angled triangle ABC, right angled at C,M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM=CM. Point D is joined to point B. Show that: △AMC≅△BMD.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7

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NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 13

In a triangle ABC, D is the mid-point of side AC such that BD = 1/2 AC. Show that ∠ABC is a right angle

Summary:

In a triangle ABC, D is the mid-point of side AC such that BD = 1/2 AC. It is shown that ∠ABC is a right angle

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