In this scale drawing, A, B and C are three corners of a park ABCD with C due north of B.

  1. (a)

    Construct the perpendicular bisector of BC.

  2. (b)

    D is on a bearing of 065° from A and equidistant from B and C.

    Mark and label point D on the drawing.

  3. (c)

    Construct the bisector of angle ABC.

  4. (d)

    Eleanor is standing inside the park ABCD such that she is closer to BC than AB and closer to B than C.

    Shade the region where Eleanor could be standing.

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Worked solution

22(a)

1. Construct the perpendicular bisector of $BC$ accurately.

Set the compass radius to more than half of BCBC. With centre BB, draw arcs on both sides of BCBC. Using the same radius with centre CC, 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 BCBC.

Answer

The perpendicular bisector of BCBC is constructed: a straight line crossing BCBC at 90∘90^\circ and passing through the midpoint of BCBC.

22(b)

1. Draw the locus satisfying the bearing condition from $A$.

From AA, draw a north line. Measure a clockwise angle of 065∘065^\circ from north and draw the ray from AA on this bearing.

2. Use the equidistant condition to locate and label $D$.

Since DD is equidistant from BB and CC, DD lies on the perpendicular bisector of BCBC constructed in part (a). Mark the intersection of the 065∘065^\circ bearing ray from AA with this perpendicular bisector and label it DD.

Answer

Point DD is marked and labelled at the intersection of the 065∘065^\circ bearing line from AA and the perpendicular bisector of BCBC.

22(c)

1. Construct the angle bisector of $\angle ABC$ accurately.

Using a compass, draw an arc centred at BB to cut BABA and BCBC. From the two intersection points, draw equal-radius arcs to meet inside ∠ABC\angle ABC. Draw a straight line from BB through the intersection of these arcs. This line is the bisector of ∠ABC\angle ABC.

Answer

The bisector of angle ABCABC is constructed: a straight line/ray from BB that divides angleABC\\angle ABC into two equal angles.

22(d)

1. Identify the two locus boundaries and choose the correct side of each boundary.

Points closer to line BCBC than to line ABAB lie on the BCBC side of the angle bisector of ∠ABC\angle ABC.

Points closer to BB than to CC lie on the BB side of the perpendicular bisector of BCBC.

Therefore, shade the intersection of these two half-regions inside the park ABCDABCD.

Answer

Shade the region inside ABCDABCD that is on the BCBC side of the angle bisector of ∠ABC\angle ABC and on the BB side of the perpendicular bisector of BCBC.

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