2.3: Constructions | Geometry Honors | Code Scholars Tutoring

How it works

A compass preserves distances and a straightedge draws lines through constructed points. The perpendicular bisector consists of points equidistant from the endpoints; an angle bisector consists of interior points equidistant from the two sides.

PA=PB\quad\text{on the perpendicular bisector of }AB

Worked examples

Example 1

Describe constructing the perpendicular bisector of $\displaystyle AB$ .

Use the same compass radius greater than half $\displaystyle AB$ from each endpoint. Join the two intersections of the arcs. This line passes through the midpoint and is perpendicular to $\displaystyle AB$ .

Example 2

Why do the intersecting arcs construction points lie on the perpendicular bisector?

Each intersection has equal distance from $\displaystyle A$ and $\displaystyle B$ because both arcs use the same radius. The equidistance locus is the perpendicular bisector.

Problems

Problem 2.3.1

For $\displaystyle A=(-3,0)$ and $\displaystyle B=(3,0)$ , give the perpendicular bisector and verify that $\displaystyle P=(0,4)$ lies on it.

Segment from negative 3,0 to 3,0, its vertical bisector, and point P at 0,4. — Every point on x = 0 is equally distant from A and B.

Show worked solution

The midpoint is $\displaystyle (0,0)$ , and the bisector is $\displaystyle x=0$ . Also $\displaystyle PA=PB=5$ .

Segment from negative 3,0 to 3,0, its vertical bisector, and point P at 0,4. The solution adds labeled relationships. — The two right triangles give PA = PB = 5.

Problem 2.3.2

Outline the construction of an angle bisector.

Show worked solution

Draw an arc centered at the vertex meeting both rays. From those two intersection points draw equal-radius intersecting arcs inside the angle. Join the vertex to their intersection.

Problem 2.3.3

Why must the radius exceed half the segment length in the perpendicular-bisector construction?

Show worked solution

Otherwise the two circles have fewer than two intersection points, so they do not determine the required line.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.3.1

Problem 2.3.2

Problem 2.3.3

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