Chapter 3 Using tools of Geometry

Lesson 3.1 Sketch – a drawing made free hand, no tools Draw – a drawing made with the tools. Compass and Straightedge Construction – a construction made using only compass and straightedge. Patty Paper Construction – a construction made using a straightedge and patty paper.

Lesson 3.2 Segment bisector – a line that passes through the midpoint of a segment.

Perpendicular Bisector – a line segment is a line that divides the line segment into two congruent parts (bisects it) and that is also perpendicular to the line segment.

C-1If a point is on the perpendicular bisector of a segment, then it is equal distance from the endpoints. (Perpendicular Bisector conjecture)

C-2If a point is equally distant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Converse of the Perpendicular Bisector Conjector)

Lesson 3.3 C-3The shortest distance from a point to a line is measured along the perpendicular from the point to the line. (Shortest Distance conjecture)

Lesson 3.4 Angle bisector – a ray that has an endpoint on the vertex of the angle and that divides the angle into two angles of equal measure.

C-4If a point is on the bisector of an angle, then it is equal distance from the sides of the angle. (Angle Bisector Conjecture)

C-5The measure of each angle of an equilateral triangle is 60 degrees.

Lesson 3.5 Parallel lines – lines that lie in the same planes and do not intersect.

Concurrent lines (segments or rays) – lines that intersect in a single point. Point of concurrency – the point that they intersect at. Lesson 3.7

C-6The three angle bisectors of a triangle are concurrent.

C-7The three perpendicular bisectors of a triangle are concurrent.

C-8 The three altitudes (or lines through the altitudes) of a triangle are concurrent.

C-9The circumcenter of a triangle is equal distance from the triangle’s three vertices. (Use the three perpendicular bisectors)

C-10The incenter of a triangle is equally distant from the triangles three sides. (Three angle bisector)

Orthocenter – the intersection of the three altitudes. Inscribed circle – circle on the inside.

Circumscribed circle – circle on the outside.

Lesson 3.8 C-11The three medians of a triangle are concurrent.

C-12The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint. Also known as the center of gravity or mass.

C-13The circumcenter, centroid, and the orthocenter are the three points of concurrency that always lie on the Euler line. C-14The centroid divides the Euler segment into two parts so that the smaller part is half as long as the larger part.