Warm Up
1) What is the difference between a
perpendicular bisector and an angle
bisector?
2) True or False: A segment bisects
another segment and creates four right
triangles.
HELP
GEOMETRY
Paper Folding Bisectors
page 272 baby blue section
share a book with your partner
1) Make a conjecture about the bisectors of the angles of a
triangle.
2) Make a conjecture about the perpendicular bisectors of the
sides of a triangle.
HELP
GEOMETRY
Concurrent- When three or more lines intersect in one point
Point of Concurrency- The point at which they intersect
3) Is there any relationship between the lengths of the point of
concurrency and the vertex or sides of the triangle?
HELP
GEOMETRY
Theorems:
The perpendicular bisectors of the sides of a triangle are
concurrent at a point equidistant from the vertices.
The bisectors of the angles of a triangle are concurrent at a
point equidistant from the sides.
(Gizmo Application )
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Find the center of the circle that circumscribes
HELP
XYZ.
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
(continued)
Vocab:
Circumcenter of a triangle
Circumscribed about a triangle
The lines y = 4 and x = 3 intersect at the point (3, 4).
This point is the center of the circle that circumscribes
HELP
XYZ.
Quick Check
GEOMETRY
Circumcenter of a Triangle- The point of
concurrency of the perpendicular bisectors of a
triangle
Circle is circumscribed about the triangle.
Incenter of a Triangle- The point of
concurrency of the angle bisectors of a triangle
We can inscribe a circle in the triangle.
HELP
GEOMETRY
Median of a Triangle
Is a segment whose endpoints are a vertex and
the midpoint of the opposite side.
Are the medians of one triangle concurrent? If
so, is there a relationship between the median
and a point of concurrency?
Centroid- Point of Concurrency of the medians
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
M is the centroid of
WOR, and WM = 16. Find WX.
The centroid is the point of concurrency of the
medians of a triangle.
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.
(Theorem 5-8)
Because M is the centroid of
2
WM = 3 WX
16 =
2
WX
3
24 = WX
HELP
2
WOR, WM = 3 WX.
Theorem 5-8
Substitute 16 for WM.
3
Multiply each side by 2 .
Quick Check
GEOMETRY
Altitude of a Triangle
is the perpendicular segment from a vertex to the line
containing the opposite side.
Theorem: The lines that contain the altitudes of a triangle are
concurrent.
In your notes, draw the altitude of a triangle for an Acute
Triangle, Right Triangle and Obtuse Triangle.
HELP
GEOMETRY
Concurrent Lines, Medians, and Altitudes
LESSON 5-3
Additional Examples
Is KX a median, an altitude, neither, or both?
Because LX = XM, point X is the midpoint of LM, and KX is a median
of KLM.
Because KX is perpendicular to LM at point X, KX is an altitude.
So KX is both a median and an altitude.
HELP
Quick Check
GEOMETRY
Practice 5-3
#1, 2, 5-11
HELP
GEOMETRY
Assignment
• P275-277
• #1-4, 9, 12-16, 19-22
HELP
GEOMETRY
Find the relationship
Create a completed triangle using straws. Try
different length straws
Find a relationship between the lengths of the
sides that complete a triangle and that would not
complete a triangle.
What conclusion can you draw about the smallest
angles and largest angles with respect to the
sides?
HELP
GEOMETRY
What do we notice…..
Triangle inequality Theorem.
The sum of any two side lengths of a
triangle is greater than the third side
length.
AB + BC > AC
A
BC + AC > AB
AC + AB > BC
HELP
B
C
GEOMETRY
Examples
Tell whether a triangle can have sides with
the given lengths. Explain.
1. 3, 5, 7
2. 4, 6, 11
HELP
GEOMETRY
Workbook
• Page 347
HELP
GEOMETRY
Comparison Property of Inequality
If a = b + c and c > 0 then a > b
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the
measure of each of its remote interior angles
m <1 > m<2 and m<1 > m<3
If two sides of a triangle are not congruent, then the larger angle lies
opposite the longer side
If two angles of a triangle are not congruent, then the longer side lies
opposite the larger angle.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the
length of the third side.
HELP
GEOMETRY
Assignment Due Monday, 11/28
• Textbook Page 292 #1-27 odd
• Try to get this done in class, ideally your
homework is to enjoy this break with
family and friends!
HELP
GEOMETRY