In a right angled triangle ABC, ∠ABC = 90°, D is mid-point of AC and AC = 13 cm. Calculate the length of BD.

6.5 cm

Reason

Draw DE || CB.

Since, DE || CB and AB is transversal,

∴ ∠AED = ∠ABC = 90° (Corresponding angles are equal)

From figure,

⇒ ∠AED + ∠DEB = 180° (Since, AB is a straight line)

⇒ 90° + ∠DEB = 180°

⇒ ∠DEB = 180° - 90° = 90°.

As, D is mid-point of AC and DE is || CB.

∴ DE bisects AB (By converse of mid-point theorem)

∴ AE = BE.

In △ AED and △ BED,

⇒ ∠AED = ∠BED [Both equal to 90°]

⇒ AE = BE [Proved above]

⇒ DE = DE [Common side]

∴ △ AED ≅ △ BED [By S.A.S. axiom]

∴ BD = AD [By C.P.C.T.C.] ………….(1)

Given,

D is mid-point of AC.

∴ AD = $\dfrac{1}{2}AC$ ……..(2)

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

⇒ BD = $\dfrac{1}{2}AC = \dfrac{1}{2} \times 13$ = 6.5 cm