Unit 1
Glossary
Geometry
An angle bisector is a line through the vertex of an angle that divides it into two congruent angles.
In this diagram, the dashed line is the angle bisector.
A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\).
To draw a circle of radius 3 units and center \(O\), use a compass to draw all the points at a distance 3 units from \(O\).
Two figures are congruent if there is a rigid motion or a sequence of rigid motions (translations, rotations, and reflections) that takes one figure onto the other.
In this figure, Triangle A is congruent to Triangle D.
- A translation takes Triangle A onto Triangle B.
- A rotation takes Triangle B onto Triangle C.
- A reflection takes Triangle C onto Triangle D.
A directed line segment is a line segment that has distance (length) and direction.
The arrow on this directed line segment shows that it starts at point \(A\) and ends at point \(B\).
An image is the result of a transformation. Every part of the original figure moves in the same way to match up with a part of the image.
This diagram shows a transformation that takes \(A\) to \(A’\).
\(A\) is the original, and \(A’\) is the image.
When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.
- A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle.
- A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.
A line of symmetry is a line that divides a figure into two parts that are mirror images of each other. When a figure is reflected across one of its lines of symmetry, it takes the figure onto itself.
These dashed lines show two lines of symmetry for a regular hexagon, and two lines of symmetry for the capital letter I.
Two lines that never intersect are called parallel. Line segments can also be parallel if they extend into parallel lines.
This figure shows two parallel line segments.
The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to the segment.
In this diagram, the dashed line is the perpendicular bisector of segment \(AB\).
A reflection is a rigid transformation that is defined by a line. It takes one point to another point that is the same distance from the given line, but on the other side. The segment from the original point to its image is perpendicular to the line of reflection.
In this figure, \(A\) is reflected across line \(m\), and \(A′\) is the image of \(A\) under the reflection.
Reflect \(A\) across line \(m\).
A figure has reflection symmetry if there is a reflection that takes the figure onto itself.
In this diagram, the letter X has reflection symmetry over each of the dashed lines.
A regular polygon is a polygon where all of the sides are congruent and all of the angles are congruent.
A rigid transformation is a move that does not change any measurements of a figure.
Translations, rotations, and reflections are rigid motions. So is any sequence of any of these.
A rotation is a rigid transformation that is defined by a center, an angle, and a direction. It takes one point on a circle to another point, using a given center. The two radii—the one from the center to the original point and the one from the center to the image—make the angle of rotation.
In this figure, \(P′\) is the image of \(P\) after a counterclockwise rotation of \(t^\circ\) using the point \(O\) as the center.
In this figure, quadrilateral \(ABCD\) is rotated \(120^\circ\) counterclockwise using the point \(D\) as the center.
A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (This does not include rotations using angles, such as \(0^\circ\) and \(360^\circ\), that take every point on a figure back to its original position.)
This hexagon has rotation symmetry 60 degrees clockwise or counterclockwise around its center.
A figure has symmetry if there is a rigid transformation that takes it onto itself. (This does not include transformations that take every point back to its original position.)
A translation is a rigid transformation that is defined by a directed line segment. It takes one point to another point so that:
- The directed line segment from the original point to its image is parallel to the given directed line segment.
- Both directed line segments have the same length and direction.
In the figure shown here, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).