# 5-4_Concurrency_of_Triangles by stariya

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```									Geometry A

Unit 5: Constructions

Lesson 4: Concurrency Properties of Triangles and It’s Applications

Objective:
 Students will be able to investigate, justify, and apply theorems about
the centroid, orthocenter, incenter, and circumcenter of a triangle.

Vocabulary:
Centroid
Orthocenter
Incenter
Circumcenter

Focus Questions:
What are the points of concurrency of a triangle? How are these points used?
Which type of triangle will have its incenter, orthocenter, circumcenter and
centroid at the same point? (right, scalene, obtuse, equilateral)

Game Plan:
Do Now
Quiz (5-1 & 5-2)
Properties of Concurrency of Triangles
Vocabulary Check
Practice
U-Try
Focus Questions

Homework: Finish U-Try
Name: ______________________                                Geometry A
Ms. Torok

Do Now:
1] Construct the following only using a straightedge and a compass.
Perpendicular at P

2] Bisect A:
Concurrency Properties of Triangles
1) Centroid
2) Incenter
3) Circumcenter
4) Orthocenter

Lines that contain the same point are called concurrent. Concurrence is the
concept of three or more lines intersecting in a single (common) point, having
a single point of intersection.

Every triangle has three perpendicular bisectors, three angle bisectors,
three medians, and three altitudes.

Etch- a- Sketch
Sketch the 3 medians:                   Sketch the 3 angle bisectors:

This point of intersection is called    This point of intersection is called
the __________                          the __________
Concurrent Lines of a Triangle
Centroid                                   Circumcenter
The medians of a triangle are              Perpendicular bisectors of sides of a
concurrent and intersect each other        triangle are concurrent at a point
in a ratio of 2:1.                         equidistant from the vertices.

Incenter                                   Orthocenter “Ortho” means “Right”
The bisectors of the angles of a           The point where the lines containing
triangle meet at a point that is equally   the altitudes are concurrent is called
distant from the sides of the              the orthocenter of a triangle.
triangle.
Vocabulary Check:
Matching:

A. Centroid          ______ the point where all the altitudes meet

B. Orthocenter       ______the point where all the medians meet

C. Circumcenter      ______the point where all the angle bisectors meet

D. Incenter          ______the point where all the perpendicular bisectors
of the sides meet

Practice using the Concurrency Properties:

I) Centroid:
Theorem: The medians of a triangle are concurrent and intersect each
other in a ratio of 2:1.

Example #1:
If point R is the centroid of triangle ABC, what is the perimeter of triangle
ABC given that segments CF, DB, and AE are equal to 2, 3 and 4 respectively?

Perimeter of triangle ABC = ________
Example #2:

In the diagram below of ΔTEM, medians      ,   , and    intersect at D, and TB
= 9. Find the length of  .

Example #3:

Point Z is the centroid of triangle ABC and CZ = 18. What is the length of
segment ?

ZF = ________
Example #4:

In the diagram of ΔABC below, Jose found centroid P by constructing the
three medians. He measured    and found it to be 6 inches.

If PF = x, which equation can be used to find x?

1. x + x = 6
2. 2x + x = 6
3. 3x + 2x = 6
2
4. x + x = 6
3

Example #5:

Point Z is the centroid of triangle ABC, ZF = 5, AD = 12 and BC = 18. What is
the perimeter of triangle DCZ?

Perimeter of triangle DCZ = ________
II) Incenter:
Theorem: The bisectors of the angles of a triangle meet at a point that
is equally distant from the sides of the triangle.

Example #1:

Which of the following points in the diagram below is the incenter of triangle
ABC?

Example #2:
Given that point D is the incenter of isosceles triangle ABC, what is the

Example #3:
Given that point S is the incenter of right triangle PQR and angle RQS is 30°,
what are the measures of angles RSQ and RPQ?

mRSQ = ________                      mRPQ = ________

III) Circumcenter:
Theorem: Perpendicular bisectors of sides of a triangle are concurrent at
a point equidistant from the vertices.

Example #1:
If circle O is circumscribed about triangle ABC, point O is what point of the
triangle?

1.   orthocenter
2.   circumcenter
3.   incenter
4.   centroid
Example #2:

The diagram below shows the construction of the center of the circle

This construction represents how to find the intersection of

1.   the angle bisectors of ΔABC
2.   the medians to the sides of ΔABC
3.   the altitudes to the sides of ΔABC
4.   the perpendicular bisectors of the sides of ΔABC

IV)       Orthocenter:
Theorem: The point where the lines containing the altitudes are
concurrent is called the orthocenter of a triangle.
Example #1:

Which point is the intersection of the altitudes of a triangle?

1] orthocenter        2] centroid           3] incenter      4] circumcenter

Example #2:

Which type of triangle would have its orthocenter on the triangle?

1] right              2] obtuse             3] scalene       4] equilateral
U-Try:

1] Which is the point of intersection of the medians of a triangle?

1.   orthocenter
2.   centroid
3.   incenter
4.   circumcenter

2] Which of the four centers always remains on or inside a triangle?

1.   incenter, only
2.   incenter and centroid
3.   orthocenter and incenter
4.   circumcenter, only

3] Three or more lines that contain the same point are called:

1.   parallel
2.   perpendicular
3.   current
4.   concurrent

4] The incenter of a triangle can be located by finding the intersection of
the:

1.   altitudes
2.   medians
3.   perpendicular bisectors of the three sides
4.   angle bisectors

5] The circumcenter of a triangle can be located by finding the intersection
of the:

1.   altitudes
2.   medians
3.   perpendicular bisectors of the three sides
4.   angle bisectors

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