Transcript Factors, Fractions, and Exponents

Warm-Up:
EOC Prep
If m<CTR = 27, what
is m<K?
A) 27
B) 54
C) 63
D) 76
Suppose RK = 8.
What is the
perimeter of TPK?
A)25
B)33
C)50
D)66
Concurrent Lines,
Medians, and
Altitudes
5.3
Today’s Goals
By the end of class today, YOU should be able to…
1. Identify and apply the properties of
medians and altitudes of a triangle.
2. Find the circumcenter of a triangle.
Concurrency
When three or more lines intersect in one
point, they are concurrent.
 The point at which they intersect is the
point of concurrency.
 For any triangle, four different sets of lines
are concurrent.

Theorems on concurrency
The perpendicular bisectors of the sides of
a triangle are concurrent at a point
equidistant from the vertices.
 The perpendicular bisectors of the sides of
a triangle are concurrent at a point
equidistant from the vertices.

Circumcenter

The point of concurrency of the
perpendicular bisectors of a triangle
Circumscribed
about/Inscribed in


A circle is circumscribed about a triangle if the
vertices of the triangle are on the circle.
A circle is inscribed in a triangle if the sides of the
triangle are tangent to the circle.
Ex.1: Circumcenters
Find the center of the circle that you can
circumscribe about OPS.
Ex.1: Solution
Two perpendicular bisectors of sides of OPS
are x = 2 and y = 3. These lines intersect at
(2, 3). This point is the center of the circle.
Incenter


The point of concurrency of the angle bisectors
of a triangle
In the following image, points X, Y, and Z are
equidistant from I, the incenter.
Median of a triangle

A segment whose endpoints are a vertex
and the midpoint of the opposite side.
Theorem 5-8

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.
Finding the median of a
triangle
D is the centroid of ABC and DE = 6.
 Find BE.
 Since D is a centroid, BD = BE and DE =
BE.

1/3 BE = DE = 6
BE = 18
Centroid of a triangle

The point of concurrency of the medians.
Altitude of a triangle

The perpendicular segment from a vertex
to the line containing the opposite side.
 Unlike
angle bisectors and medians, an
altitude of a triangle can be a side of a
triangle or it may lie outside the triangle.
Orthocenter of a triangle

The point of intersection of the lines
containing the altitudes of the triangle.
Theorem 5-9

The lines that contain the altitudes of a
triangle are concurrent.
Homework
 Page
259 #s 1, 4, 8, 9, 10, 12, 16
 Page 260 #s 19-22, 27
 The
assignment can also be found at:
• http://www.pearsonsuccessnet.com/snpapp/iText/produc
ts/0-13-037878-X/Ch05/05-03/PH_Geom_ch0503_Ex.pdf