Chapter 17: Review

The chapter problem, solved

The garden is $24$ m east-west and $18$ m north-south. At scale $1$ cm to $3$ m, draw a rectangle $8$ cm by $6$ cm. Its centre is located by the perpendicular bisectors of opposite sides. A circle of radius $4$ m around the centre marks the planting bed; on this plan its radius is $4/3$ cm.

The path is the diagonal. Its real length is

\sqrt{24^2+18^2}=\sqrt{900}=30\text{ m}.

So it is exactly $30$ m, not shorter or longer. The angle east of north is $\tan^{-1}(24/18)\approx 53.1^\circ$ , so the bearing is approximately $053^\circ$ . Label the plan's sides and vertices using the shape words from 17.1; use a plan and elevations if the garden includes a raised structure.

Mixed practice: choose a method and explain why

For each item, name the method first. Then work it out and give a reason for your choice.

A triangular prism has triangular ends. How many rectangular faces does it have? Which shape property tells you?
A cuboid measures $7$ cm by $4$ cm by $3$ cm. State its plan and front elevation dimensions.
Construct the perpendicular bisector of a $7$ cm segment. Explain why each point on your line is equally far from the endpoints.
Describe the locus of points $3$ cm from point $P$ and the locus of points $2$ cm from a straight boundary.
Explain why the perpendicular from a point to a line gives the shortest distance.
A map scale is $1$ cm to $4$ km. Work out the real distance represented by $6.5$ cm.
Write the three-figure bearing for south-east.
Earlier chapter link: a rectangular room is $4.8$ m by $3.6$ m. Use a scale of $1$ cm to $0.6$ m to find its plan dimensions, then find the diagonal on the plan.

Answers and checks

Three rectangular faces, one for each side of the triangular end.
Plan $7$ cm by $4$ cm; front elevation $7$ cm by $3$ cm.
Equal-radius arcs from both endpoints cross on the perpendicular bisector. Each crossing is equally far from both endpoints.
A circle centred at $P$ with radius $3$ cm; two lines parallel to the boundary, each $2$ cm away.
Any sloping segment forms the hypotenuse of a right triangle, so it is longer than the perpendicular side.
$6.5\times4=26$ km.
$135^\circ$ .
The plan is $8$ cm by $6$ cm; its diagonal is $10$ cm.

What you can now do

describe and draw common 2D and 3D shapes using conventional terms;
read and construct plans and elevations;
perform ruler-and-compass constructions and use loci;
use bearings and scale drawings to describe measured routes.

Your relevance reflection

Return to your example from the opener. Add one way a shape property, construction, locus, plan or bearing could help with it.