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5.3 Medians and Altitudes of a Triangle

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Find the distance from vertex K to midpoint N. The distance from K(5, 2) to N (5, ... The distance formula isn't going to disappear any time soon. ... – PowerPoint PPT presentation

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Title: 5.3 Medians and Altitudes of a Triangle


1
5.3 Medians and Altitudes of a Triangle
  • Geometry
  • Mrs. Spitz
  • Fall 2004

2
Objectives
  • Use properties of medians of a triangle
  • Use properties of altitudes of a triangle

3
Assignment
  • pp. 282-283 1-11, 17-20, 24-26

4
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.

5
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

6
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.

7
CENTROIDS -
ALWAYS INSIDE THE TRIANGLE
8
Medians
  • The medians of a triangle have a special
    concurrency property as described in Theorem 5.7.
    Exercises 13-16 ask you to use paper folding to
    demonstrate the relationships in this theorem.

9
THEOREM 5.7 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

10
So what?
  • The centroid of a triangle can be used as its
    balancing point. Lets try it. Ive handed out
    triangle to each and every one of you. Construct
    the medians of the triangles in order to great
    the centroid in the middle. Then use your pencil
    to balance your triangle. If it doesnt balance,
    you didnt construct it correctly.

11
Ex. 1 Using the Centroid of a Triangle
  • P is the centroid of ?QRS shown below and PT 5.
    Find RT and RP.

12
Ex. 1 Using the Centroid of a Triangle
  • Because 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.

13
Ex. 2 Finding the Centroid of a Triangle
  • Find 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.

14
Ex. 2 Finding the Centroid of a Triangle
  • The coordinates of N are
  • 37 , 610 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.

15
Ex. 2 Finding the Centroid of a Triangle
  • Determine 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, 24), or
    (5, 6).

16
Distance Formula
  • Ive told you before. The distance formula isnt
    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.

17
Objective 2 Using altitudes of a triangle
  • An 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.

18
Ex. 3 Drawing Altitudes and Orthocenters
  • Where is the orthocenter located in each type of
    triangle?
  • Acute triangle
  • Right triangle
  • Obtuse triangle

19
Acute Triangle - Orthocenter
?ABC is an acute triangle. The three altitudes
intersect at G, a point INSIDE the triangle.
20
Right Triangle - Orthocenter
?KLM is a right triangle. The two legs, LM and
KM, are also altitudes. They intersect at the
triangles right angle. This implies that the
ortho center is ON the triangle at M, the vertex
of the right angle of the triangle.
21
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.
22
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.

23
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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