4.8

Concurrent Lines

Notes(Vocab) Altitude:

is the line segment from a vertex of a triangle perpendicular to the opposite side.

Altitudes

Notes(Vocab) Orthocenter:

is the intersection of the altitudes of the triangle.

Acute Triangle - Orthocenter B A D G F E ∆ABC is an acute triangle. The three altitudes intersect at G, a point INSIDE the triangle.

E

Right Triangle - Orthocenter K J ∆KLM is a right triangle. The two legs, LM and KM, are also altitudes.

They intersect at the triangle’s right angle. This implies that the ortho center is ON the triangle at M, the vertex of the right angle of the triangle.

M L

Obtuse Triangle - Orthocenter ∆YPR is an obtuse triangle. The three lines that contain the altitudes intersect at W, a point that is OUTSIDE the triangle.

P Z Y Q W X R

Notes(Vocab)

Median:

is the segment drawn from a vertex of a triangle to the midpoint of the opposite side.

Medians of a triangle A A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. For instance in ∆ABC, shown at the right, D is the midpoint of side BC. So, AD is a median of the triangle B D MEDIAN C

Notes(Vocab)

Centroid:

is the intersection of the medians and is known as the “

center of mass

”.

(Also known as the balancing point)

Centroids of the Triangle The three medians of a triangle are concurrent (they meet). The point of concurrency is called the CENTROID OF THE TRIANGLE. The centroid, labeled P in the diagrams in the next few slides are ALWAYS inside the triangle. acute triangle P CENTROID

CENTROIDS P centroid RIGHT TRIANGLE ALWAYS INSIDE THE TRIANGLE P obtuse triangle centroid

Notes(Vocab)

Perpendicular Bisector:

is the line or segment that passes through the midpoint of a side and is perpendicular to the side.

Perpendicular Bisector of a Triangle • A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

Perpendicular Bisector

Notes(Vocab)

Circumcenter:

is the center of a circumscribed circle made by the intersections of the perpendicular bisectors.

About concurrency • The three 90° Angle Right Triangle perpendicular bisectors of a triangle are concurrent. The point of concurrency may be

inside

the triangle,

on

the triangle, or

outside

the triangle.

A B C

About concurrency • The three Acute Angle Acute Scalene perpendicular Triangle bisectors of a triangle are concurrent. The point of concurrency may be

inside

the triangle,

on

the triangle, or

outside

the triangle.

About concurrency • The three perpendicular bisectors of a triangle Obtuse Angle Obtuse Scalene Triangle are concurrent. The point of concurrency may be

inside

the triangle,

on

the triangle, or

outside

the triangle.

Notes(Vocab) Angle Bisector:

is the line, segment or ray that bisects an angle of the triangle.

Intersection of Angle Bisectors

Notes(Vocab)

Incenter:

is the center of an inscribed circle. Made by the intersection of the angle bisectors.

Notes(Vocab) Inscribed Circle:

is a circle that is inside of a triangle and touches all three sides. (The center is the intersection of the angle bisectors in the triangle, known as the incenter)

Notes(Vocab)

Circumscribed Circle:

is a circle outside of the triangle touching all three vertices.

(The center is the intersection of the perpendicular bisectors known as the cirumcenter)

• When three or more concurrent lines (or rays or segments) intersect in the same point, then they are called concurrent lines (or rays or segments). The point of intersection of the lines is called the point of concurrency .