Question

in a triangle, if square of one side is equal to the sum of other two sides, then prove that the angle opposite to the first side is right angle.​

Answer 1

Step-by-step explanation:

Given :

ABC is a triangle

AC². = AB² + BC²

to prove :

angle B = 90°

construction :

construct a ∆PQR. Right angled at Q such that , PQ = AB and QR = BC

Proof :

In ∆PQR

PR² = PQ² + QR²( By Pythagoras theorem)

PR² = AB² + BC² ----1 (AB=PQ and QR= BC)

AC² = AB² + BC² ----2 (Given)

From equation (1) and (2) , we have

AC². = PR²

AC. =. PR ----3

now in ∆ABC and ∆PQR

AB = PQ

BC = QR

AC = PR [from 3]

∆ ABC congruent to ∆PQR [ By SSS congruency ]

Therefore by CPCT

ang. B = ang. Q

ang. Q = 90°

ang. B = 90°

---------------HENCE PROVED---------------