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!