In △ ABC, D Is the Mid - Point of Side AC Such That BD =        AC .  Show That  ∠ABC = 90.

By Team Aakash Byju's | 29th January 2023

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Let's draw a triangle ABC where we are not sure if the measure of ∠ABC is 90° or not. Let us join B with a point D which is a mid-point of AC.

Since it is given that BD=

Consider ΔABD, where AD = BD, so the angles opposite to these sides are equal. i.e., ∠ABD = ∠BAD

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we can say BD = AD = CD.

Similarly, in ΔBDC, where  CD = BD, so the angles opposite to these sides are equal. i.e., ∠CBD = ∠BCD

Finally, since AD = CD = AD  => ∠ABD = ∠BAD = ∠CBD = ∠BCD = x (say).

From the angle sum property of a triangle, ∠ABC + ∠BCA + ∠CAB = 180° (∠ABD + ∠DBC) + ∠BCA + ∠CAB  = 180°

(x + x) + x + x = 180°

(∵ ∠BCA is as same as ∠BCD, and ∠CAB is as same as ∠BAD)

4x = 180° or x = 45°

Since x = 45°, ∠ABD = ∠BAD = ∠CBD = ∠BCD = 45°

Also, ∠ABC = ∠ABD + ∠DBC ∠ABC = 45° + 45° = 90°

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