Refer to the diagram below. QR and QP have already been drawn.

(a)
In the space below, construct a quadrilateral PQRS such that PS = 7 cm, angle QRS = $110^\circ$ and angle PSR is an acute angle.

QR and QP have already been drawn.

Answer
(b)
Construct the perpendicular bisector of PQ.
(c)
The perpendicular bisector in (b) intersects the line QR at T. Measure the angle QTP.

Answer ___ $^\circ$

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Worked solution

20(a)

1. Construct the line on which $S$ must lie using the given angle at $R$.

At $R$ , construct an angle of $110^\circ$ with arm $RQ$ . Draw the ray from $R$ on the required side of $QR$ so that
$\angle QRS=110^\circ$ .
Point $S$ must lie on this ray.

2. Locate $S$ using the given length $PS=7\text{ cm}$ and complete the quadrilateral.

With centre $P$ and radius $7\text{ cm}$ , draw an arc to intersect the ray from $R$ .
Choose the intersection point $S$ for which $\angle PSR$ is acute.
Join $P$ to $S$ and $R$ to $S$ to complete quadrilateral $PQRS$ .

Answer

Quadrilateral $PQRS$ constructed, with $PS=7\text{ cm}$ , $\angle QRS=110^\circ$ and $\angle PSR$ acute.

20(b)

1. Construct equal arcs from $P$ and $Q$ and draw the perpendicular bisector.

Set the compass radius to more than half of $PQ$ . With centre $P$ , draw arcs on both sides of $PQ$ . Using the same radius and centre $Q$ , draw arcs to intersect the first pair of arcs. Join the two arc-intersection points with a straight line. This line is the perpendicular bisector of $PQ$ .

Answer

The perpendicular bisector of $PQ$ is constructed.

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Refer to the diagram below. QR and QP have already been drawn.

Worked solution

20(a)

20(b)

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