The diagram shows a circle and a chord AB. B is due east of A.

(a)
Locate and label a point C on the circle such that $\angle ABC = 110^\circ$ .
(b)
The point D is equidistant from A and B. It is also given that $\angle ADC = 70^\circ$ and that D is above AB. Locate and label the point D.
(c)
By drawing suitable lines, locate and label the center of the circle as O.
(d)
Measure and write down the bearing of B from O.

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Worked solution

23(a)

1. Construct the required angle at B and locate C on the circumference

At $B$ , place the protractor with its baseline along $BA$ . Mark $110^\circ$ on the side above $AB$ , then draw the ray from $B$ through this mark. Label the point where this ray meets the circle as $C$ .

Thus $\angle ABC = 110^\circ$ .

Answer

Point $C$ is labelled at the intersection of the circle and the ray from $B$ such that $\angle ABC = 110^\circ$ .

23(b)

1. Construct the locus of points equidistant from $A$ and $B$

Draw the perpendicular bisector of $AB$ . Since $D$ is equidistant from $A$ and $B$ , $D$ must lie on this perpendicular bisector.

2. Locate the point on this locus that gives $\angle ADC=70^\circ$ and is above $AB$

Using the constructed point $C$ , find the point on the perpendicular bisector of $AB$ such that the angle between $DA$ and $DC$ is $70^\circ$ . Select the intersection above $AB$ and label it $D$ .

Answer

Point $D$ is labelled at the intersection above $AB$ of the perpendicular bisector of $AB$ and the ray/arc position satisfying $\angle ADC=70^\circ$ .

23(c)

1. Locate the centre using perpendicular bisectors of chords

Draw the perpendicular bisector of chord $AB$ and the perpendicular bisector of another chord, e.g. $AC$ . Their point of intersection is the centre of the circle. Label this point $O$ .

Answer

$O$ is labelled at the intersection of the perpendicular bisectors of two chords of the circle, e.g. $AB$ and $AC$ .

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The diagram shows a circle and a chord AB. B is due east of A.

Worked solution

23(a)

23(b)

23(c)

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