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Welcome To
2
Bisectors, Medians, and Altitudes
Inequalities and Triangles
2 Triangles Inequalities
The Triangle Inequality
Indirect Proof
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Bisectors, Medians, and Altitudesfor 100

• Define orthocenter

4
Answer

• Orthocenter The intersection point of the
  altitudes of a triangle.

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5
Bisectors, Medians, and Altitudes for 200

• Where can the perpendicular bisectors of the
  sides of a right triangle intersect?

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Answer

• On the triangle.

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7
Bisectors, Medians, and Altitudes for 300

• Where is the center of the largest circle that
  you could draw inside a given triangle? What is
  the special name for this point?

8
Answer

• The intersection of the angle bisectors of a
  triangle the point is called the incenter.

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9
Bisectors, Medians, and Altitudes for 400

• Find the center of the circle that you can
  circumscribe about the triangle.

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Answer
The circumcenter is made by the perpendicular
bisectors of a triangle. Only need to find
the Intersection of 2 lines Median of AB is (-3,
½) Perp Line y 1/2 Median of BC is (-1,
½) Perp Line x -1 Cicumcenter (-1, 1/2)
A
B
C
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11
Bisectors, Medians, and Altitudes for 500

• In triangle ACE, G is the centroid and AD 12.
  Find AG and GD.

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Answer
The centroid divides the medians of a triangle
into parts of length (2/3) and (1/3) so, AG
(2/3)(AD) (2/3)(12) 8 GD (1/3)(AD)
(1/3)(12) 4
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13
Inequalities and Triangles for 100

• Define Comparison Property

14
Answer

• For all real numbers a, b
• altb, ab, or agtb

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Inequalities and Triangles for 200

• Define Inequality

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Answer

• For any real numbers a and b, agtb iff there is a
  positive number c such that a b c

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17
Inequalities and Triangles for 300

• If in triangle ABC, AB 10,
• BC 12 and CA 9, which angle has the greatest
  measure?

18
Answer

• Angle A has the greatest measure because it is
  opposite side BC, which is the longest side.

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19
Inequalities and Triangles for 400

• If in triangle ABC, ltA 10 degrees, ltB 85
  degrees and ltC 85 degrees, which side is the
  longest?

20
Answer
Side AC and Side AB are the longest because they
are opposite the largest angles (85 degrees).
Since there are two equal angles, the triangle is
isosceles.
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21
Inequalities and Triangles for 500

• Define the exterior angle inequality theorem

22
Answer
If an angle is the exterior angle of a triangle,
then its measure is greater than the measure of
either of its corresponding remote interior angles
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23
Indirect Proof for 100

• Define Indirect Reasoning

24
Answer
Indirect reasoning reasoning that assumes the
conclusion is false and then shows that this
assumption leads to a contradiction.
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25
Indirect Proof for 200

• List the three steps for writing an indirect
  proof

26
Answer

• List the three steps for writing an indirect
  proof
• Assume that the conclusion is false
• Show that this assumption leads to a
  contradiction of the hypothesis, or some other
  fact, such as a definition, postulate, theorem,
  or corollary
• Point out that because the false conclusion leads
  to an incorrect statement, the original
  conclusion must be true

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27
Indirect Proof for 300

• Prove that there is no greatest even integer.

28
Answer

• Assume that there is a greatest even integer, p.
• Then let p2 m
• mgtp and p can be written 2x for some integer x
  since it is even. Then
• p2 m 2x2 m 2(x1) m. x 1 is an
  integer, so 2(x1) means m is even. Thus m is an
  even number and mgtp
• Contradiction against assuming p is the greatest
  even number

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29
Indirect Proof for 400

• Prove that the negative of any irrational number
  is also irrational.

30
Answer

• Assume x is an irrational number, but -x is
  rational.
• Then -x can be written in the form p/q where p,q
  are integers and q does not equal 0,1.
• x -(p/q) -p/q -p and q are integers and
  thus -p/q is a rational number
• Contradiction with x is irrational

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31
Indirect Proof for 500

• Given Bobby and Kina together hit at least 30
  home runs. Bobby hit 18 home runs.
• Prove Kina hit at least 12 home runs.

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Answer

• Assume Kina hit fewer than 12 home runs. This
  means Bobby and Kina combined to hit at most 29
  home runs because Kina would have hit at most 11
  home runs and Bobby hit 18, so 1118 29. This
  contradicts the given information that Bobby and
  Kina together hit at least 30 home runs.
• The assumption is false. Therefore, Kina hit at
  least 12 home runs.

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33
The Triangle Inequalityfor 100
Write the triangle inequality theorem
34
Answer
The sum of the lengths of any two sides of a
triangle is greater than the length of the third
side.
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35
The Triangle Inequalityfor 200

• The shortest segment from a point to a line
  is_______

36
Answer
The segement perpendicular to the line that
passes through the point.
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37
The Triangle Inequalityfor 300

• Can the following lengths be sides of a triangle?
• 4, 5, 9

38
Answer
No, 45 9, in order to be a triangle 45 gt 9
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39
The Triangle Inequalityfor 400

• Determine the range for the measure of the third
  side or a triangle give that the measures of the
  other two sides are 37 and 43

40
Answer
43 37 6 43 37 80 So the range for the
third side, x, is 6 lt x lt 80
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41
The Triangle Inequalityfor 500
Prove that the perpendicular segment from a point
to a line is the shortest segment from the point
to the line
P
1
2
3
l
A
B
42
Answer
Statements Reasons
PA - l PB is any non-perpendicular segment from P to l Given
lt1 and lt2 are right angles - lines form right angles
lt1 is congruent to lt2 All right angles are congruent
mlt1 mlt2 Def. of Congruent angles
mlt1 gt mlt3 Exterior angle inequality theorem
mlt2 gt mlt3 Substitution
PBgt PA If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is lover than the side opposite the lesser angle
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43
2 Triangles Inequalitiesfor 100

• Write out the SAS Inequality theorem

44
Answer

• If two sides of a triangle are congruent to two
  sides of another triangle, and the included angle
  in one triangle has a greater measure than the
  included angle in the other, then the third side
  of the first triangle is longer than the third
  side of the second triangle.

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45
2 Triangles Inequalitiesfor 200

• Write out the SSS Inequality theorem

46
Answer

• If two sides of a triangle are congruent to two
  sides of another triangle, and the third side in
  one triangle is longer than the third side in the
  other, then the angle between the pair of
  congruent sides in the first triangle is greater
  than the corresponding angle in the second
  triangle.

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47
2 Triangles Inequalitiesfor 300

• Given ST PQ, SR QR and ST 2/3 SP
• Prove mltSRP gt mltPRQ

Q
R
T
P
S
48
Answer
Statements Reasons
SR QR ST PQ ST 2/3 SP SP gt ST Given
PR PR Reflexive
SP gt PQ Substitution
mltSRP gt m lt PRQ SSS Inequality
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49
2 Triangles Inequalitiesfor 400

• Given KL JH JK HL
• mltJKH mltHKL lt mltJHK mltKHL
• Prove JH lt KL

K
J
H
L
50
Answer
Statements Reasons
mltJKH mltHKL lt mltJHK mltKHL JK HL KL JH Given
mltHKL m lt JHK Alt. Interior Angle Theorem
mltJKH mltJHK lt mltJHK mltKHL Substitution
mltJKH lt mlt KHL Subtraction
HK HK Reflexive
JH lt KL SAS Inequality Theorem
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51
2 Triangles Inequalitiesfor 500

• Given PQ is congruent to SQ
• Prove PR gt SR

S
P
T
R
Q
52
Answer
Statements Reasons
PQ is congruent to SQ Given
QR QR Reflexive Property
mltPQR mltPQS mltSQR Angle Addition Postulate
mltPQR gt mlt SQR Definition of Inequality
PR gt SR SAS Inequality Theorem
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