Lesson 5-1
Unit 5
Relationships in Triangles
Day 1/2 5-1 Centroid and Orthocenter
HW: 5-H1
Day 3 5-2 Incenter and Circumcenter
HW: 5-H2
Day 4 5-3 Inequalities and Triangles
HW: 5-H3
Day 5 5-4 Exterior Angle Inequality Theorem
HW: 5-H4
Day 6 Quiz
Day 7 5-5 Triangle Inequality Theorem
HW: 5-H5
Day 8/9 Unit 5 Cumulative Review
Day 10/11 Unit 5 Test
5-1 Centroid and Orthocenter
Define:
a. Concurrent Lines-
b. Point of Concurrency-
CENTROID
To draw a centroid: draw a segment from each vertex to midpoint of each opposite side (aka: a median)
The medians of a triangle are concurrent. Notice that the point of
concurrence (CENTROID) is in the interior of the triangles.
KEY IDEA
The centroid divides the medians into a ______ ratio.
Examples
1. Z is the centroid of triangle ABC. BD = 8, ZF = 3, and AZ = 10. Find:
a) CZ = ____ c) BC = ____
b) DZ = ____ d) AD = ____
2. In the diagram, M is the centroid of [pic], CM = 36, MQ = 30, and TS = 56.
Find the length of AM and RM.
3. In (ABC, [pic] are medians, and P is the centroid.
B
a. Find x if DP = 4x -3 and CP =30
b. Find y if AP = y and EP =18
c. Find z if FP = 2z + 4 and BP = 42
ORTHOCENTER
To draw an orthocenter: draw a segment from each vertex, perpendicularly to the opposite side (aka: a altitude)
The altitudes of a triangle are concurrent. Notice that the point of concurrence (ORTHOCENTER) is NOT ALWAYS in the interior of the triangles.
KEY IDEA
Ortho means ________ and altitudes always
form right angles
5-2 Incenter and Circumcenter
Incenter
To draw an incenter: draw a segment, that BISECTS each vertex, to the opposite side
(aka: angle bisector)
The angle bisectors of a triangle are concurrent. Notice that the point of concurrence (INCENTER) is ALWAYS in the interior of the triangles.
KEY IDEA
The incenter is ___________ from each side of a triangle
Examples:
1. In the diagram, N is the incenter of [pic]. Which statement cannot be determined from the given information?
(1) [pic]
(2) [pic]
(3) [pic]
(4) [pic]
2. Point D is the incenter of triangle XYZ. Find:
AD = ______
BZ = ______
Circumcenter
To draw an circumcenter: draw a segment that is perpendicular to and bisects each side (aka: perpendicular bisector)
The perpendicular bisectors of a triangle are concurrent. Notice that the point of concurrence (CIRCUMCENTER) is NOT ALWAYS in the interior of the triangles.
KEY IDEA
Any point on the perpendicular bisector is ___________ from the endpoints.
Example:
1. Point G is the circumcenter of triangle ABC. Find:
AD = ______
GC = ______
AF = ______ (simplest radical form)
Regents Level Practice
1. 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?
[1] scalene
[2] isosceles
[3] equilateral
[4] right
2. The centroid of a triangle is the intersection of the ________________.
[1] medians
[2] angle bisectors
[3] perpendicular bisectors
[4] altitudes
3. 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?
[1] inside the triangle
[2] on the triangle
[3] outside the triangle
4. Is it possible to locate the incenter
of every triangle? Why?
[pic]
3. Inequalities and Triangles
Side Relationships
If one side of a triangle is _______________ than another side, then the ____________ opposite that ____________ is larger than the _____________ opposite the shorter side.
Symbols:
Angle Relationships
If one angle of a triangle is _______________ than another angle, then the _________ opposite that _______________ is larger than the _____________opposite the smaller angle.
Symbols:
Examples:
1. Name the angles from largest to smallest.
2. Name the sides from shortest to longest.
3. In (ABC, m[pic]A = 95, m[pic]B = 50, and m[pic]C = 35. Which expression correctly relates the lengths of the sides of this triangle. *Hint: Draw a picture
(1) AB < BC < CA
(2) AB < AC < BC
(3) AC < BC < AB
(4) BC < AC < AB
4. In triangle DEF, DE = 12, EF = 14 and DF= 20. Write the angles as an inequality.
5. If (RST, m[pic]R = 71( and m[pic]S = 37(, then which is true?
(1) ST > RS
(2) RS = ST
(3) RS > RT
(4) RT > ST
6. In isosceles triangle DEF, vertex angle D measures 80(. What conclusions can be made about the sides of the triangle?
7. In (ABC, m[pic]C = 55( and m[pic]C > m[pic]B. Which is the longest side of the
triangle?
8. In (ABC, m[pic]B > m[pic]C and m[pic]C > m[pic]A. Which side of the triangle is the longest? Which is the shortest?
4. Exterior Angle Inequality Theorem
Exterior Angle Inequality Theorem
If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles.
Examples:
1. In the diagram below of (ABC with side [pic] extended through D, m[pic]A = 37( and m[pic]BCD = 117(. Which side of (ABC is the longest side? Justify your answer.
B
A
C D
2. Determine the angle with the greatest measure, for each set of
angles.
(1) [pic]1, [pic]3, [pic]4
(2) [pic]4, [pic]8, [pic]9
(3) [pic]2, [pic]3, [pic]7
(4) [pic]7, [pic]8, [pic]10
3. Side [pic] of (PQR is extended through Q to T. Which statement is not always true?
(1) m[pic]RQT > m[pic]R (2) m[pic]RQT = m[pic]P + m[pic]R
(3) m[pic]RQT > m[pic]P (4) m[pic]RQT > m[pic]PQR
4. In (ABC, [pic] is extended to E, and D is a point on [pic]. Which is the following is true?
(1) m[pic]ADE > m[pic]ABC (2) m[pic]ADE = m[pic]ABC
(3) m[pic]ACB = m[pic]ABC (4) m[pic]ABC > m[pic]ADE
A
E C D B
5. Use the Exterior Angle Inequality Theorem to
list all of the angles that satisfy the stated condition.
a. all angles whose measures are less than m(2
b. all angles whose measures are greater than m(3
5. Triangle Inequality Theorem
Triangle Inequality Theorem: _________________________________________________
___________________________________________________________________________
Therefore:
B
4 6
A 7 C
Identify Sides of a Triangle
Determine whether the given measures can be the lengths of the sides of a triangle.
a. 3, 6, 8
b. 7, 9, 18
c. 6, 8, 10
d. 2, 2, 4
e. 5, 5, 5
Determining Sides of a Triangle
When given two sides of a triangle to determine the possible lengths of the third side, the third side must be______________________________
_________________________________________________________.
Examples:
1. In (QRS, QR = 12 and RS = 8. Which measure cannot be QS?
(1) 10 (2) 9 (3) 3 (4) 16
2. What is the range for the lengths of the third side of a triangle that
has 2 sides of 4 and 11?
3. What is the range for the lengths of the third side of a triangle that
has 2 sides of 6 and 19?
4. What is the range for the lengths of the third side of a triangle that
has 2 sides of 7 and 29?
5. In the diagram below of (ABC, D is a point on [pic]. AC = 10, CB=20, AD = 4.
C
The length of DB could be
(1) 5
(2) 16
(3) 26
(4) 30
-----------------------
P
C
F
A
E
D
B
A
C
B
A
C
B
A
C
8
16
4
B
A
C
57(
86(
37(
.
A
B
D
................
................
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