Lesson 5-1Lesson 5-1Bisectors, Medians, Bisectors, Medians,

and Altitudesand Altitudes

Ohio Content Standards:

Ohio Content Standards:

• Formally define geometric figures.

Ohio Content Standards:• Formally define and explain key aspects of

geometric figures, including:a. interior and exterior angles of polygons;b. segments related to triangles (median, altitude, midsegment);c. points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);

Perpendicular Bisector

Perpendicular Bisector

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

Theorem 5.1

Theorem 5.1

Any point on the perpendicular bisector of a segment is equidistant from

the endpoints of the segment.

Example

C D

A

B

. and

then, bisects and If

BDBCADAC

CDABCDAB

Theorem 5.2

Theorem 5.2

Any point equidistant from the endpoints of a segment

lies on the perpendicular bisector of the segment.

Example

C D

A

B

. ofbisector lar perpendicu on the lies then , If

. ofbisector lar perpendicu on the lies then ,C If

CDBBDBC

CDAADA

Concurrent Lines

Concurrent Lines

When three or more lines intersect at a common point.

Point of Concurrency

Point of Concurrency

The point of intersection where three or more lines

meet.

Circumcenter

Circumcenter

The point of concurrency of the perpendicular bisectors

of a triangle.

Theorem 5.3Circumcenter Theorem

Theorem 5.3Circumcenter Theorem

The circumcenter of a triangle is equidistant from the vertices of the triangle.

Example

CA

B

.then

, ofer circumcent theis If

CKBKAK

ABCK

circumcenter

K

Theorem 5.4

Theorem 5.4Any point on the angle

bisector is equidistant from the sides of the angle.

A C

B

Theorem 5.5

Theorem 5.5Any point equidistant from

the sides of an angle lies on the angle bisector.

A

B

C

Incenter

Incenter

The point of concurrency of the angle bisectors.

Theorem 5.6Incenter Theorem

Theorem 5.6Incenter Theorem

The incenter of a triangle is equidistant from each side of

the triangle.

CA

Bincenter

KP

Q

R

Theorem 5.6Incenter Theorem

The incenter of a triangle is equidistant from each side of

the triangle.

CA

Bincenter

KP

Q

R

If K is the incenter of ABC, then KP = KQ

= KR.

Median

Median

A segment whose endpoints are a vertex of a triangle and

the midpoint of the side opposite the vertex.

Centroid

Centroid

The point of concurrency for the medians of a triangle.

Theorem 5.7Centroid Theorem

Theorem 5.7Centroid Theorem

The centroid of a triangle is located two-thirds of the

distance from a vertex to the midpoint of the side opposite

the vertex on a median.

Example

CA

B

D L E

F

centroid

.3

2 and ,

3

2

,3

2 , of centroid theis If

CDCLBFBL

AEALABCL

Altitude

Altitude

A segment from a vertex in a triangle to the line

containing the opposite side and perpendicular to the line

containing that side.

Orthocenter

Orthocenter

The intersection point of the altitudes of a triangle.

Example

CA

B

D

L

E

F

orthocenter

Points U, V, and W are the midpoints of YZ, ZX, and XY, respectively. Find a, b, and c.

Y

W U

XV Z

7.45c 8.7

15.22a

3b + 2

The vertices of QRS are Q(4, 6), R(7, 2), and S(1, 2). Find

the coordinates of the orthocenter of QRS.

Assignment:

Pgs. 243-245 13-20 all