Geometry
Constructions
Lesson 3-4: Concurrence in triangles
Objective:
 Students will be able to identify the 4 concurrences in triangles by the
construction.
 Students will apply the construction skills to concurrence
constructions.
Procedure:
 Go over HW!
 Quiz
 Paper folding: concurrences (optional)
 Notes/Practice: concurrence
 http://www.mathopenref.com/triangle
 Student reference notes on concurrences(separate doc)
Homework: HWS 3-4 concurrences
Name:________________________
Geometry
Date:_______________
Quiz: 3-1_copy/bisect
1. Using your compass and straightedge, construct equilateral triangle using the given segment as
the length of its base. (2 pts.)
2. Using your compass and straightedge, construct a segment with length x + z – y. Label it CD
(3 pts.)
X
y
z
3. Using your compass and straightedge, copy angle ABC. (3 pts.)
4. Bisect the given angle. (3 pts.)
5. Construct the perpendicular bisector of PQ . (2 pts.)
6)
(2 pts.)
Lesson 3-4: Concurrence in triangles
Definition of concurrence: 3 or more line intersect at a common point.
Location, Location, Location:
Use the following visual data to find the patterns of the concurrences.
Centroid:
Acute
Right
Obtuse
Orthocenter:
Acute
Right
Obtuse
Incenter:
Acute
Right
Obtuse
Circumcenter:
Acute
Right
Obtuse
Record the location of each:
acute
right
obtuse
Centroid
Orthocenter
Incenter
Circumcenter
All Of : Altitudes – orthocenter
My Children: medians - centroid
Always Bite Into: angle bisectors - incenter
Peanut Butter Cookies: perpendicular bisectors - circumcenter
Matching:
___________
the point where all the altitudes meet
___________
the point where all the medians meet
___________
the point where all the angle bisectors meet
____________
the point where all the perpendicular bisectors of the sides meet
---------------------------------------------------------------------------------------------------------------------Draw a sketch of an inscribed circle.
You would need to construct
the ___________________
to find the incenter.
Draw a sketch of a circumscribed circle.
You would need to construct
the ___________________
to find the circumcenter.
Think about it!
In which type of triangle would the incenter, circumcenter and centroid all occur at the same
point? ____________________________
Construction: Inscribe a circle in a triangle
Steps:
1) Construct angle bisectors to locate the incenter.
2) Drop a perpendicular from incenter to opposite side if the triangle.
3) The segment- where the perpendicular line intersects with the side to the
incenter is the radius of the circle.
4) Construct the circle with center at the incenter and the given radius.
Theorems about concurrence in triangles
Sketch each theorem:
Theorem: Circumcenter
Perpendicular bisectors of sides of a triangle are concurrent at a point equidistant
from the vertices.
Theorem: Incenter
The bisectors of the angles of a triangle meet at a point that is equally distant
from the sides of the triangle.
Theorem: Centroid
The medians of a triangle are concurrent and intersect each other in a ratio of 2:1.
Euler line:
Circumcenter O
Centroid G
Orthocenter H
HG:GO is a 2:1 ratio.
Geometry Lesson 3-4: Concurrence in triangles
Circumscribe a circle about the triangle below.
Short Answer:
1) The perpendicular bisectors of  ABC meet at point G. If
= 12,
= 6, and
= 3, find
.
2) The governor wants to build a new library for three cities at the intersection of the medians of XYZ .
If the length of an incoming road (XA) is 14 miles, find the distance from the library to the midpoint of
X
YZ.
B
A
D
Y
C
Z
3) For which of the following, the point of concurrency always lie inside the triangle?
1. Medians
2. Altitudes 3. Interior angular bisectors 4. Perpendicular bisectors
A ) 1, 2, 3 & 4
B ) 1 & 3 only
C ) 1 & 2 only
D ) 2 & 4 only
4) M is the centroid of ABC . If BE=48, find BM and ME.
5) M is the centroid of ABC . If BM =12, find ME and BE.
6) Which type of triangle would have its orthocenter outside the triangle?
1. right
2.obtuse
3.scalene
4.equilateral
7) Which type of triangle would have its orthocenter on the triangle?
1. right
2.obtuse
3.scalene
4.equilateral
8) Which type of triangle will have its incenter, orthocenter, circumcenter and centroid at the
same point?
1. right
2.obtuse
3.scalene
4.equilateral
9) 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.
Name:_______________________________
Geometry 3-4 HWS
Date:___________
1) The circumcenter of an acute triangle is located inside the triangle.
The circumcenter of an obtuse triangle is located outside the triangle.
Where is the circumcenter of a right triangle located in relation to the triangle?
2) Scalene triangle ABC is shown below. The distance from the centroid of a triangle to the
circumcenter is 8 units. How far is the centroid from the orthocenter?
3) You are looking at a triangle where the orthocenter, the centroid and the circumcenter are all
the same point. What type of triangle are you looking at?
4) Finding Lengths of Medians: D is the centroid of △ABC and DA 8.
Find DF and AF.
DF ________ AF ______
5) If QC = 5 inches QS= 10 in, SR= 12 in. and QR= 18 in.
Find each:
a) RC=___
b) SC=___
c) perimeter of triangle SCR=_____
6)
Circle the correct response
7) If TI = 12 and XT = 4, find XI, YI and IZ.
8) The measures of angles A and B of parallelogram ABCD are in the ratio of 4:5.
Find the measure of angle D.
9) Given: Quadrilateral TALK with vertices T(-5, 4), A (0, 4) L(0, -1) K (-5, -1)
Show that the diagonals of Quadrilateral TALK bisect each other.