Question

PQRS is a square and angle ABC = 900

as shown in the figure. If AP = BQ = CR, then

prove that angle BAC = 450

Answer 1

Answer:

Given that:- ABC is a △ with ∠BAC=90° and PQRS is a square.

To prove:- RS

2

=BR×SC

Proof:-

In △APQ and △RBP,

∠APQ=∠PBR[∵corresponding angles]

∠PAQ=∠BRP[∵each 90°]

∴△APQ∼△RBP[AA criteria].....(1)

Similarly, in △AQP and △SCQ

∠AQP=∠QCS[∵corresponding angles]

∠PAQ=∠CSQ[∵each 90°]

△AQP∼△SCQ[AA criteria].....(2)

From eq

n

(1)&(2), we get

△RBP∼△SCQ

As we know that corresponding sides of similar triangles are proportional.

∴

SQ

BR

=

SC

RP

⇒BR×SC=RP×SQ.....(3)

As given that PQRS is a square.

∴PQ=QS=RS=PR.....(4)

From eq

n

(3)&(4), we get

RS×RS=RP×SQ

⇒RS

2

=RP×PQ

Hence proved that RS

2

=RP×PQ.