Let A B be a diameter of a circle ω and let C be a point on ω , different from A and B . The perpendicular from C intersects A B at D and ω at E(≠ C) . The circle with centre at C and radius C D intersects ω at P and Q

Correct answer: 08 C P E Q is cyclic quadrilateral aligned & C P × E Q+C Q × P E=C E × P Q & r(E Q+P E)=2 r P Q & 2 P Q=E Q+P E & P Q+E Q+P E=24 & P Q+2 P Q=24 & P Q=8 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Circle

Olympiad workbook · IOQM · Circle

Let A B be a diameter of a circle ω and let C be a point on ω ,...

Let

A B

be a diameter of a circle

\omega

and let

C

be a point on

\omega

, different from

A

and

B

. The perpendicular from

C

intersects

A B

D

and

\omega

E(\neq \mathrm{C})

. The circle with centre at

C

and radius

C D

intersects

\omega

P

and

Q

. If the perimeter of the triangle

P E Q

is 24 , find the length of the side

P Q

Let A B be a diameter of a circle ω and let C be a point on ω , different from A and B . The perpendicular from C

Correct answer

Step-by-step solution

C P E Q

is cyclic quadrilateral

\begin{aligned} & C P \times E Q+C Q \times P E=C E \times P Q \\ & r(E Q+P E)=2 r P Q \\ & 2 P Q=E Q+P E \\ & P Q+E Q+P E=24 \\ & P Q+2 P Q=24 \\ & \therefore \quad P Q=8 \end{aligned}

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