The diagram shows a line segment PQ.

(i)
Construct the perpendicular bisector of PQ, showing your constructions clearly.

Answer
(ii)
Explain clearly why your construction method gives the perpendicular bisector of PQ.

Answer

Bisectors Construction 中文

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Worked solution

12(i)

1. Draw equal-radius arcs from both endpoints and join their intersections

Set the compass radius to more than half of $PQ$ .

With centre $P$ , draw arcs above and below $PQ$ .

Using the same radius and centre $Q$ , draw arcs to intersect the first pair of arcs at two points.

Draw a straight line through the two intersection points. This line is the perpendicular bisector of $PQ$ .

Answer

The perpendicular bisector of $PQ$ is the straight line drawn through the two intersection points of the equal-radius arcs above and below $PQ$ .

12(ii)

1. State why each arc intersection point is equidistant from $P$ and $Q$

Let the two intersection points of the arcs be $A$ and $B$ .

Since the arcs drawn with centres $P$ and $Q$ have the same radius,
$AP=AQ$ and $BP=BQ$ .

2. Apply the perpendicular bisector property to justify the constructed line

A point equidistant from $P$ and $Q$ lies on the perpendicular bisector of $PQ$ .

Since both $A$ and $B$ are equidistant from $P$ and $Q$ , the line through $A$ and $B$ is the perpendicular bisector of $PQ$ .

Answer

Let the two arc intersection points be $A$ and $B$ . Since the arcs drawn from $P$ and $Q$ have the same radius, $AP=AQ$ and $BP=BQ$ . Hence $A$ and $B$ are both equidistant from $P$ and $Q$ , so they lie on the perpendicular bisector of $PQ$ . Therefore the line $AB$ is the perpendicular bisector of $PQ$ .

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The diagram shows a line segment PQ.

Worked solution

12(i)

12(ii)

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