Daily Warm-Up Quiz

1. Which of your classmates disclosed to a teacher that Mrs. M. sometimes refers to Makenna as Mackenzie…and vice versa?

2. Who told this same teacher that period 2 Geometry is my “favorite class”? How did you determine this?

3. Who shared that since Monday, Kaylin has been renamed “Kylin”?

Mrs. McConaughy Geometry 1

Mrs. McConaughy Geometry 2

Relationships in Triangles

Concurrent Lines, Medians and Altitudes

Mrs. McConaughy Geometry 3

Part I: Identifying Properties of Angle

Bisectors and Perpendicular Bisectors

in Triangles

Mrs. McConaughy Geometry 4

In this lesson, we will identify

properties of perpendicular bisectors and angle bisectors in

triangles.∆ OPS

Mrs. McConaughy Geometry 5

Long before the first pencil and paper, some curious person drew a triangle in the sand and bisected the three angles. He noted that the bisectors met in a single point and decided to repeat the experiment on an extremely obtuse triangle. Again, the bisectors concurred. Astonished, the person drew yet a third triangle, and the same thing happened yet again! Unlike squares and circles, triangles have many centers. The ancient Greeks found four: incenter, centroid, circumcenter, and orthocenter. Triangle Centers: http://faculty.evansville.edu/ck6/tcenters/index.html

Mrs. McConaughy Geometry 6

Vocabulary & Key Concepts

When three or more lines intersect in one

point, they are called _____________.The point at which they intersect is

called the _________________.

concurrent

point of concurrency

Mrs. McConaughy Geometry 7

Vocabulary and Key Concepts

THEOREM: The bisectors of the angles of a ∆ are concurrent at a point (incenter) equidistant from the sides.

The point of concurrency of the angle bisectors of a triangle is called the _________ of the triangle.

incenter

I is the incenter of the ∆.

Mrs. McConaughy Geometry 8

Checking for Understanding

City planners want to locate a fountain equidistant from three straight roads that enclose a park. Explain how they can find the location.

Andover Road

Mariposa

Boulevard

Hig

hway

101

Check your solution here!

Mrs. McConaughy Geometry 9

Vocabulary and Key Concepts

The point of concurrency of the perpendicular bisectors of a triangle is called the ____________ of the triangle.

THEOREM: The perpendicular bisectors of the angles of a ∆ are concurrent at

a point (circumcenter) equidistant from the vertices.

circumcenter

Alert! The common distance is the radius of a circle that passes through the vertices.

O is the circumcenter.

Mrs. McConaughy Geometry 10

Checking for Understanding

Find the center of the circle that you can circumscribe about ∆ OPS.

Solution:

Checking for Understanding: Finding the Circumcenter

Two perpendicular bisectors of the sides of ∆ OPS are x = 2 and y = 3. These lines intersect at (2,3). This point is the center of the circle.

Mrs. McConaughy Geometry 11

Homework

Mrs. McConaughy Geometry 12

Part II: Identifying Properties of Medians and Altitudes in Triangles

Mrs. McConaughy Geometry 13

In this lesson, we will

identify properties of medians

and altitudes in triangles.

∆ OPS

Mrs. McConaughy Geometry 14

Median of a Triangle

A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.

Vertex

Midpoint

Mrs. McConaughy Geometry 15

Vocabulary and Key Concepts The point of

concurrency of the medians of a triangle is called the___________ of the triangle.

centroid

G is the centroid.

Theorem: The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.

FG = 2/3 FC EG = 2/3 EB AG = 2/3 AD

Mrs. McConaughy Geometry 16

Checking for Understanding

G is the centroid of ∆ ABC and DG = 6. Find AG.

Finding the Lengths of Medians.

G is the centroid.

AG = 2/3 AD;

DG = 1/3 AD

6 = 1/3 AD

18 = AD

Mrs. McConaughy Geometry 17

Altitude of a Triangle:An altitude of a triangle is the

segment from a vertex to the line containing the opposite side.

Unlike angle bisectors and medians, an altitude can lie inside, on, or outside the triangle.

Acute Triangle: Interior Altitude

Right Triangle: Altitude is a side

Obtuse Triangle: Exterior Altitude

perpendicular

Mrs. McConaughy Geometry 18

Altitude of a Triangle

The lines containing the altitudes of a triangle are concurrent at the orthocenter.

Theorem: The lines that contain the altitudes of a triangle are concurrent.

http://www.mathopenref.com/triangleorthocenter.html

Mrs. McConaughy Geometry 19

Identifying Medians and Altitudes

A

H

B

C

M

Is CM a median, altitude, or neither? Explain.

Is BH a median, altitude, or neither? Explain.

Mrs. McConaughy Geometry 20

Homework

Mrs. McConaughy Geometry 21

Solution: City Planning Dilemma

The roads form a triangle around the park. By our new theorem, we know that the __________________ of a triangle are concurrent at a point

_________ from the sides. The city planners should find the point of concurrency of the _______________

of the triangle formed and locate the fountain there.

bisectors of the angles

equidistant

bisectors of the angles

Click here to return to the lesson!