5.3 Medians and Altitudes of a Triangle

G

F

D

E

B

A E

P. Spitz – Taos H.S. 2

Objectives:• Identify properties of

perpendicular bisectors and angle bisector

• Use properties of medians and altitudes of a triangle

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• The perpendicular bisectors of a triangle intersect at a common point.

• When three or more lines intersect at a common point the lines are called concurrent lines.

• Their point of intersection is called the point of concurrency.

• The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter

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Circumcenter Theorem

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

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Example 1

• Find the center of the circle that circumscribes triangle XYZ

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Using Medians of a Triangle

In Lesson 5.2, you studied two types of segments of a triangle: perpendicular bisectors of the sides and angle bisectors. In this lesson, you will study two other types of special types of segments of a triangle: medians and altitudes.

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Medians of a triangle

A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. For instance in ∆ABC, shown at the right, D is the midpoint of side BC. So, AD is a median of the triangle

MEDIAN

D

A

B

C

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Centroids of the Triangle

The three medians of a triangle are concurrent (they meet). The point of concurrency is called the CENTROID OF THE TRIANGLE. The centroid, labeled P in the diagrams in the next few slides are ALWAYS inside the triangle.

CENTROID

acute triangle

P

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CENTROIDS -

centroid

RIGHT TRIANGLE

Pcentroid

obtuse triangle

P

ALWAYS INSIDE THE TRIANGLE

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Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

If P is the centroid of ∆ABC, then

AP = 2/3 AD, BP = 2/3 BF, and CP = 2/3 CE

PE

D

F

B

A

C

13

Ex. 1: Using the Centroid of a TriangleP is the centroid of

∆QRS shown below and PT = 5. Find RT and RP.

P

T

R

Q

S

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Ex. 1: Using the Centroid of a TriangleBecause P is the centroid.

RP = 2/3 RT.

Then PT= RT – RP = 1/3 RT. Substituting 5 for PT, 5 = 1/3 RT, so

RT = 15.

Then RP = 2/3 RT = 2/3 (15) = 10► So, RP = 10, and RT

= 15.

P

T

R

Q

S

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Ex. 2: Finding the Centroid of a TriangleFind the coordinates of the

centroid of ∆JKL

You know that the centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

Choose the median KN. Find the coordinates of N, the midpoint of JL.

P

N

J (7, 10)

M

K (5, 2)

L (3, 6)

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Ex. 2: Finding the Centroid of a TriangleThe coordinates of N are:3+7 , 6+10 = 10 , 16 2 2 2 2

Or (5, 8)

Find the distance from vertex K to midpoint N. The distance from K(5, 2) to N (5, 8) is 8-2 or 6 units.

P

N

J (7, 10)

M

K (5, 2)

L (3, 6)

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Ex. 2: Finding the Centroid of a TriangleDetermine the

coordinates of the centroid, which is 2/3 ∙ 6 or 4 units up from vertex K along median KN.

►The coordinates of centroid P are (5, 2+4), or (5, 6).

P

N

J (7, 10)

M

K (5, 2)

L (3, 6)

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Distance Formula

I’ve told you before. The distance formula isn’t going to disappear any time soon. Exercises 21-23 ask you to use the Distance Formula to confirm that the distance from vertex J to the centroid P in Example 2 is two thirds of the distance from J to M, the midpoing of the opposite side.

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Objective 2: Using altitudes of a triangleAn altitude of a triangle is the

perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on, or outside the triangle. Every triangle has 3 altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle.

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Ex. 3: Drawing Altitudes and Orthocenters• Where is the orthocenter located in

each type of triangle?

a. Acute triangle

b. Right triangle

c. Obtuse triangle

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Acute Triangle - Orthocenter

G

F

D

E

B

A E

∆ABC is an acute triangle. The three altitudes intersect at G, a point INSIDE thetriangle.

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Right Triangle - Orthocenter

J

K

M L

∆KLM is a right triangle. The twolegs, LM and KM, are also altitudes.They intersect at the triangle’s rightangle. This implies that the orthocenter is ON the triangle at M, thevertex of the right angle of thetriangle.

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Obtuse Triangle - Orthocenter∆YPR is an obtuse triangle. The three lines that contain the altitudes intersect at W, a point that is OUTSIDE the triangle.

QW Y

P

R

Z

X

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Theorem 5.8 Concurrency of Altitudes of a triangle

The lines containing the altitudes of a triangle are concurrent.

If AE, BF, and CD are altitudes of ∆ABC, then the lines AE, BF, and CD intersect at some point H.

H

EA

C

BF

D

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FYI --

Exercises 24-26 ask you to use construction to verify Theorem 5.8. A proof appears on pg. 838 for your edification . . .

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Assignment

• pp. 282-283 #1-11, 17-20, 24-26