Integrated Math 1 / Unit 9: Coordinate Geometry and Constructions
9.6 More Constructions
Notes
Compass-and-straightedge constructions rely on equal distances.
- Equilateral triangle: draw segment $AB$. With radius $AB$ draw an arc from $A$ and an arc from $B$. The crossing point $C$ is the third vertex, since $AC = BC = AB$. Each angle is $60^\circ$.
- A point on the perpendicular bisector of $\overline{AB}$ is equidistant from $A$ and $B$.
- Parallel line through a point $P$: copy an angle so that corresponding angles are congruent.
- Perpendicular through a point: draw arcs from the point that cut the line twice, then bisect that segment.
Practice
An equilateral triangle is constructed on segment $AB$. Why are all three sides equal?
The arcs from $A$ and $B$ have radius $AB$, and $C$ lies on both arcs.
Point $P$ lies on the perpendicular bisector of $\overline{AB}$ and $PA = 9$ cm. What is $PB$?
Every point on the perpendicular bisector is equidistant from the endpoints.
To construct a line through point $P$ parallel to line $\ell$, which idea is used?
Lines cut by a transversal are parallel when corresponding angles are congruent.
Ready for more? Take the Unit 9 test.