Q. 64.0( 17 Votes )

# Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is an isosceles triangle.

Answer :

Given: Bisector of ∠BAC of ΔABC is perpendicular to side BC

To Prove: ΔABC is an isosceles triangle.

Proof:

In ΔABD and ΔACD

Since, AD is the angle Bisector of ΔABC

∴ ∠BAD = ∠CAD

AD = AD ……….[Common Side]

∠ADB = ∠ADC ……[Both equal to 90°]

So, by ASA congruency test

ΔABD ≅ ΔACD

Therefore,

AB = AC ………………. corresponding sides of congruent triangles.

∠ABD = ∠ACD ……………… corresponding angles of congruent triangles.

∴ ∠ABC = ∠ACB

Since, AB = AC and ∠ABC = ∠ACB so, ΔABC is an isosceles triangle.

Rate this question :

In figure 3.25, ∠P ≅ ∠R seg PQ ≅ seg RQ

Prove that, ΔPQT ≅ ΔPQS

MHB - Math Part-II

ΔABC is isosceles in which AB = AC. seg BD and seg CE are medians. Show that BD = CE.

MHB - Math Part-IIIn figure 3.59, point D and E are on side BC of ΔABD, such that BD = CE and AD = AE. Show that ΔABD ≅ ΔACE.

MHB - Math Part-II

Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is an isosceles triangle.

MHB - Math Part-IIIn figure 3.51, in ΔABC, seg AD and seg BE are altitudes and AE = BD.

Prove that seg AD ≅ seg BE

MHB - Math Part-II

Prove that an equilateral triangle is equiangular

MHB - Math Part-II