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Bisectors,
Medians, and
Altitudes
Inequalities
and Triangles
Indirect Proof
The Triangle
Inequality
2 Triangles &
Inequalities
$100 $100 $100 $100 $100
$200 $200 $200 $200 $200
$300 $300 $300 $300 $300
$400 $400 $400 $400 $400
$500 $500 $500 $500 $500
Bisectors, Medians, and Altitudes
for $100
Define: orthocenter
Answer
Orthocenter –The
intersection point of the
altitudes of a triangle.
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Bisectors, Medians, and Altitudes for
$200
Where can the perpendicular
bisectors of the sides of a right
triangle intersect?
Answer
On the triangle.
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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?
Answer
The intersection of the
angle bisectors of a
triangle; the point is
called the incenter.
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Bisectors, Medians, and Altitudes for
$400
Find the center of the circle that you can
circumscribe about the triangle.
Answer
The circumcenter is made by the
perpendicular bisectors of a triangle.
Only need to find the
Intersection of 2 lines:
A
Median of AB is (-3, ½)
Perp Line: y = 1/2
Median of BC is (-1, ½)
B
C
Perp Line: x = -1
Cicumcenter: (-1, 1/2)
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Bisectors, Medians, and Altitudes for
$500
In triangle ACE, G is the centroid and
AD = 12. Find AG and GD.
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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Inequalities and Triangles for $100
Define: Comparison Property
Answer
For all real numbers a, b:
a<b, a=b, or a>b
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Inequalities and Triangles for $200
Define: Inequality
Answer
For any real numbers a and b, a>b iff
there is a positive number c such
that a = b + c
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Inequalities and Triangles for $300
If in triangle ABC, AB = 10,
BC = 12 and CA = 9, which
angle has the greatest
measure?
Answer
Angle A has the greatest
measure because it is opposite
side BC, which is the longest
side.
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Inequalities and Triangles for $400
If in triangle ABC, <A = 10
degrees, <B = 85 degrees and
<C = 85 degrees, which side
is the longest?
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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Inequalities and Triangles for $500
Define the exterior angle
inequality theorem
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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Indirect Proof for $100
Define: Indirect Reasoning
Answer
Indirect reasoning – reasoning that
assumes the conclusion is false
and then shows that this
assumption leads to a
contradiction.
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Indirect Proof for $200
List the three steps for writing an
indirect proof:
Answer
List the three steps for writing an
indirect proof:
1) Assume that the conclusion is false
2) Show that this assumption leads to a
contradiction of the hypothesis, or
some other fact, such as a definition,
postulate, theorem, or corollary
3) Point out that because the false
conclusion leads to an incorrect
statement, the original conclusion
must be true
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Indirect Proof for $300
Prove that there is no greatest
even integer.
Answer
Assume that there is a greatest even
integer, p.
Then let p+2 = m
m>p and p can be written 2x for some
integer x since it is even. Then:
p+2 = m; 2x+2 = m; 2(x+1) = m. x+ 1 is an
integer, so 2(x+1) means m is even.
Thus m is an even number and m>p
Contradiction against assuming p is the
greatest even number
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Indirect Proof for $400
Prove that the negative of any
irrational number is also
irrational.
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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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.
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 11+18 = 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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The Triangle Inequality
for $100
Write the triangle
inequality theorem:
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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The Triangle Inequality
for $200
The shortest segment from a
point to a line is_______
Answer
The segement perpendicular
to the line that passes
through the point.
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The Triangle Inequality
for $300
Can the following lengths
be sides of a triangle?
4, 5, 9
Answer
No, 4+5 = 9, in order to
be a triangle 4+5 > 9
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The Triangle Inequality
for $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:
Answer
43 – 37 = 6
43 + 37 = 80
So the range for the third side, x, is:
6 < x < 80
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The Triangle Inequality
for $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
A
3
l
B
Answer
Statements
Reasons
PA ┴ l
PB is any non-perpendicular segment
from P to l
Given
<1 and <2 are right angles
┴ lines form right angles
<1 is congruent to <2
All right angles are congruent
m<1 = m<2
Def. of Congruent angles
m<1 > m<3
Exterior angle inequality theorem
m<2 > m<3
Substitution
PB> 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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2 Triangles & Inequalities
for $100
Write out the SAS Inequality
theorem
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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2 Triangles & Inequalities
for $200
Write out the SSS Inequality
theorem
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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2 Triangles & Inequalities
for $300
Given: ST = PQ, SR = QR and
ST = 2/3 SP
Prove: m<SRP > m<PRQ
Q
R
T
S
P
Answer
Statements
Reasons
SR = QR
ST = PQ
ST = 2/3 SP; SP > ST
PR = PR
SP > PQ
m<SRP > m < PRQ
Given
Reflexive
Substitution
SSS Inequality
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2 Triangles & Inequalities
for $400
Given: KL || JH; JK = HL;
m<JKH + m<HKL < m<JHK + m<KHL
Prove: JH < KL
K
J
H
L
Answer
Statements
Reasons
m<JKH + m<HKL <
m<JHK + m<KHL
JK = HL
KL || JH
Given
m<HKL = m < JHK
m<JKH + m<JHK <
m<JHK + m<KHL
Alt. Interior Angle Theorem
Substitution
m<JKH < m< KHL
Subtraction
HK = HK
JH < KL
Reflexive
SAS Inequality Theorem
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2 Triangles & Inequalities
for $500
Given: PQ is congruent to SQ
Prove: PR > SR
S
P
T
Q
R
Answer
Statements
Reasons
PQ is congruent to SQ
QR = QR
Given
Reflexive Property
m<PQR = m<PQS + m<SQR Angle Addition Postulate
m<PQR > m< SQR
PR > SR
Definition of Inequality
SAS Inequality Theorem
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