4.8 Concurrent Lines

4.84.8

Concurrent LinesConcurrent Lines

Notes(Vocab) Altitude:Altitude: is the line segment from

a vertex of a triangle perpendicular to the opposite side.

AltitudesAltitudes

Notes(Vocab) Orthocenter:Orthocenter: is the intersection

of the altitudes of the triangle.

Acute Triangle - Orthocenter

G

F

D

E

B

A E

∆ABC is an acute triangle. The three altitudes intersect at G, a point INSIDE thetriangle.

Right Triangle - Orthocenter

J

K

M L

∆KLM is a right triangle. The twolegs, LM and KM, are also altitudes.They intersect at the triangle’s rightangle. This implies that the orthocenter is ON the triangle at M, thevertex of the right angle of thetriangle.

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.

QW Y

P

R

Z

X

Notes(Vocab) Median:Median: is the segment drawn

from a vertex of a triangle to the midpoint of the opposite side.

Medians of a triangle

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

MEDIAN

D

A

B

C

Notes(Vocab) Centroid: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.

CENTROID

acute triangle

P

CENTROIDS -

centroid

RIGHT TRIANGLE

Pcentroid

obtuse triangle

P

ALWAYS INSIDE THE TRIANGLE

Notes(Vocab) Perpendicular Bisector: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 TrianglePerpendicular 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:Circumcenter: is the center of a

circumscribed circle made by the intersections of the perpendicular bisectors.

About concurrency

• The three 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

90° Angle-Right Triangle

About concurrency

• The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle.

Acute Angle-Acute Scalene Triangle

About concurrency

• The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle.

Obtuse Angle-Obtuse Scalene Triangle

Notes(Vocab) Angle Bisector:Angle Bisector: is the line,

segment or ray that bisects an angle of the triangle.

Intersection of Angle BisectorsIntersection of Angle Bisectors

Notes(Vocab) Incenter:Incenter: is the center of an

inscribed circle. Made by the intersection of the angle bisectors.

Notes(Vocab)Inscribed Circle: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: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.