Question

in an isosceles triangle ABC the base BC is produced both the ways to P and Q such that PC into BQ is equals to AC^2 prove that triangle PCA similar to triangle ABQ question 98

Answer 1

SOLUTION :  

Given : ΔABC is isosceles ∆ and AP x BQ = AC²

To prove : ΔAPC∼ΔBCQ.

Proof :  

ΔABC is an isosceles triangle AC = BC.

AP x BQ = AC²  (given)

AP x BQ = AC x AC

AP x BQ = AC x BC

AP/BC = ABQ……….(1).

Since, AC = BC

Then, ∠CAB = ∠CBA    

(angles opposite to equal sides are EQUAL)

180° – ∠CAP = 180° – ∠CBQ

∠CAP = ∠CBQ ………..(2)

In ∆APC &  ΔBCQ

AP/BC = AC/BQ [From equation 1]

∠CAP = ∠CBQ [From equation 2]

ΔAPC∼ΔBCQ  (By SAS similarity criterion)

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