Vocabulary and Definitions
What does it mean to bisect a segment or angle?
To divide it into two congruent parts.
What two tools are used in classical geometric constructions?
A compass and an unmarked straightedge.
In a diagram, line m crosses segment AB at a right angle exactly halfway between A and B. What type of line is m?
The perpendicular bisector of AB.
What does it mean to bisect an angle?
To divide it into two congruent angles.
If A is the center of a circle and B is on the circle, what is AB?
a radius
Define a circle in your own words.
The set of all points equidistant from a center point.
What is the first step in nearly every construction?
Set the compass width and draw arcs from specific points.
If CD is the perpendicular bisector of AB, what can you say about CD's relation to A and B?
Any point on CD is equidistant from A and B.
Describe how to construct an angle bisector using a compass.
Draw an arc across both rays, draw arcs from intersection points that cross, then connect the vertex to that intersection.
If two circles share the same radius, what can you conclude about them?
They are congruent circles.
Define a radius and a diameter, and describe their relationship.
Radius = distance from center to circle;
diameter = 2 × radius.
Describe the steps to construct a perpendicular bisector of a segment.
Draw equal-radius arcs from both endpoints; connect the arc intersections.
If a point P is on the perpendicular bisector of AB, what is true about PA and PB?
They are congruent.
If P is on the bisector of ∠ABC, what can you say about P’s distance to each side?
They’re equal; P is equidistant from both sides.
Explain how constructing overlapping circles can prove two segments are equal.
Intersections form congruent radii, showing equal lengths.
What does it mean when a point is equidistant from two other points?
It’s the same distance from both, lying on their perpendicular bisector.
How do you construct a line perpendicular to a given line through a point not on the line?
Draw arcs from the point that cross the line, then arcs from those intersection points that cross above; connect intersections.
When you construct the perpendicular bisector of a segment, why must the arcs drawn from both endpoints have the same radius?
So that their intersection points are equidistant from both endpoints, ensuring the bisector is exactly in the middle.
How can you prove triangles are isosceles using a circle construction?
If two vertices lie on the same circle with a shared center, the radii to them are congruent.
When completing a complex construction, why is it important to keep the compass width the same between steps?
To preserve congruent radii and ensure accuracy.
Explain what a perpendicular line is and how you can identify one in a diagram.
It intersects another line at 90°.
Describe how to construct a regular hexagon inside a circle.
Use the radius as the side length, step the compass around the circle six times, then connect the points.
If AB is bisected by CD at M, what must be true about AM, MB, and ∠CMB?
AM = MB and ∠CMB = 90°
Two circles centered at A and B intersect at C and D. What type of quadrilateral is ABCD?
A rhombus (four congruent sides).
In a figure with circles centered at A, B, and C where AB = BC, what can you conclude about triangle ABC?
It’s isosceles with equal sides AB and BC.