CH-19-001 · Circle theorem chain with an exterior angle

A, B and C lie on a circle with centre O and radius 22 cm. B is on the major arc AC. Angle ABC = 63°. OA is extended beyond A to D.

Diagram not accurately drawn

(a) Find angle DAC. Give a reason for each stage of your working.

(b) Find the length of the minor arc AC in terms of π.

Part (a) Circle Theorems (Maths | Reason Table) & Arc Length Booklet p. 38, p. 85 & p. 86

Method

Write each calculation beside the geometric reason that justifies it

Worked Example (CH-19-001 · Part (a))

Find angle DAC. Give a reason for each stage of your working.

Maths	Reason
$\angle AOC=126^\circ$	Angle at the centre is twice the angle at the circumference on the same arc.
$OA=OC$	Radii of the same circle are equal, so triangle AOC is isosceles.
$\angle OAC=\frac{180-126}2=27^\circ$	The remaining two angles in the triangle are equal.
$\angle DAC=180-(27)=153^\circ$	OA and AD form a straight line.

Ans = 153°

Part (b) Circle Theorems (Maths | Reason Table) & Arc Length Booklet p. 38, p. 85 & p. 86

Method

1) Angle at centre AOC = 2 × ABC

2) Isosceles triangle OAC (radii) ⇒ OAC = (180 − AOC)/2

3) Straight line DAC = 180 − OAC

4) Arc = (AOC/

0) × 2πr

Worked Example (CH-19-001 · Part (b))

Find the length of the minor arc AC in terms of π.

\frac{126}{360}\times 2\pi\times 22

\frac{77}{5}\pi

Ans =

\frac{77}{5}\pi