In the scale drawing, PQRS represents a plot of land, which is to be used for a park. Scale 1 cm to 0.5 km

  1. (a)

    Find the actual distance between P and Q.

    Ans: ___ km

  2. (b)

    Measure the bearing of P from S.

    Ans: ___ °

  3. (c)

    Construct the angle bisector of ∠PQR.

  4. (d)

    Construct the perpendicular bisector of RS.

  5. (e)

    A water fountain is to be built in the park, nearer to R than to S and equidistant to PQ and QR. Mark a possible location for the water fountain to be built and label it as F.

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Worked solution

24(c)

1. Construct the angle bisector of $\angle PQR$ using standard compass arcs.

With centre QQ, draw an arc cutting QPQP and QRQR at two points.
Using the same compass radius from these two points, draw two arcs inside ∠PQR\angle PQR so that they intersect.
Draw a straight line from QQ through the intersection of the two arcs. This line is the angle bisector of ∠PQR\angle PQR.

Answer

The angle bisector of ∠PQR\angle PQR is constructed as a straight line from QQ dividing ∠PQR\angle PQR into two equal angles.

24(d)

1. Construct the perpendicular bisector of $RS$ using equal-radius compass arcs.

Set the compass radius to more than half of RSRS. With centre RR, draw arcs above and below RSRS. Using the same radius and centre SS, draw arcs to intersect the first pair of arcs. Draw a straight line through the two arc-intersection points. This line is the perpendicular bisector of RSRS.

Answer

The perpendicular bisector of RSRS is constructed as a straight line crossing RSRS at its midpoint at 90∘90^\circ.

24(e)

1. Identify and mark a point satisfying both locus conditions.

Since FF is equidistant from PQPQ and QRQR, FF must lie on the angle bisector of ∠PQR\angle PQR.

Since FF is nearer to RR than to SS, FF must be on the same side of the perpendicular bisector of RSRS as RR.

Choose any point inside the park PQRSPQRS that lies on the angle bisector of ∠PQR\angle PQR and is on the RR-side of the perpendicular bisector of RSRS. Label this point FF.

Answer

Mark FF anywhere inside PQRSPQRS on the angle bisector of ∠PQR\angle PQR, on the side of the perpendicular bisector of RSRS nearer to RR.

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