Question

in an isosceles triangle the base AB produced both ways in P and Q such that triangle ACP similar to triangle BCQ

Answer 1

Hey mate,
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)

HOPE THIS ANSWER WILL HELP YOU❤️❤️

Answer 2

here is the answer.......