三角形花园 PQR 的规划图满足 $QR = 9.5\text{ cm}$ 且 $\angle PQR = 30^\circ$ 。PQ 已为你画出。

(a)
在提供的空间中作三角形 PQR。
(b)
1. (i)
  作线段 RP 的垂直平分线。
2. (ii)
  作 $\angle QPR$ 的角平分线。
(c)
需要在花园内建造一个长椅 S，使得它到 PR 的距离小于到 PQ 的距离，且 $SR \ge 3\text{ cm}$ 。阴影表示可能建造 S 的区域。

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完整解法

21(a)

1. Construct the direction of $QR$ using the given angle at $Q$.

At $Q$ , use a protractor to draw a ray making $30^\circ$ with $QP$ on the required side of the given line $PQ$ . This ray is the possible direction of $QR$ .

2. Locate $R$ and complete the triangle.

On the $30^\circ$ ray, measure $9.5\text{ cm}$ from $Q$ and mark this point as $R$ , so that $QR=9.5\text{ cm}$ . Join $R$ to $P$ using a ruler.

答案

Triangle $PQR$ constructed with $\angle PQR=30^\circ$ and $QR=9.5\text{ cm}$ .

21(b)(i)

1. Construct equal arcs from $R$ and $P$ to locate two points equidistant from $R$ and $P$.

Set the compass radius to more than half of $RP$ . With centre $R$ , draw arcs above and below $RP$ . Using the same radius and centre $P$ , draw arcs to intersect the first pair of arcs.

2. Draw the perpendicular bisector through the arc intersections.

Join the two arc intersection points with a straight line. This line is the perpendicular bisector of $RP$ .

答案

The perpendicular bisector of $RP$ is constructed.

21(b)(ii)

1. Mark two equal-distance points on the arms of $\angle QPR$.

With centre $P$ , draw an arc cutting $PQ$ and $PR$ at two points.

2. Locate a point equidistant from the two arms and draw the angle bisector.

Using the same compass radius, draw arcs from the two points marked on $PQ$ and $PR$ so that the arcs intersect inside $\angle QPR$ . Draw a straight line from $P$ through this intersection point.

答案

The bisector of $\angle QPR$ is constructed.

21(c)

1. Identify the half of the triangle where points are nearer to $PR$ than to $PQ$.

The bisector of $\angle QPR$ was constructed in part (b)(ii). Points on this bisector are equidistant from $PR$ and $PQ$ . Therefore, points nearer to $PR$ than to $PQ$ are in the half of the triangle on the $PR$ side of this angle bisector.

2. Apply the distance condition $SR \ge 3\text{ cm}$ and shade the required intersection region.

Draw a circle or arc with centre $R$ and radius $3\text{ cm}$ . Since $SR \ge 3\text{ cm}$ , the possible positions of $S$ must be outside or on this circle. Shade only the region that is:

inside triangle $PQR$ ,
on the $PR$ side of the angle bisector of $\angle QPR$ ,
outside or on the circle centred at $R$ with radius $3\text{ cm}$ .

答案

Shade the part inside triangle $PQR$ that is on the $PR$ side of the bisector of $\angle QPR$ and outside or on the circle with centre $R$ and radius $3\text{ cm}$ .

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三角形花园 PQR 的规划图满足 QR=9.5 cmQR = 9.5\text{ cm}QR=9.5 cm 且 ∠PQR=30∘\angle PQR = 30^\circ∠PQR=30∘。PQ 已为你画出。