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Welcome To. Bisectors, Medians, and Altitudes. Inequalities and Triangles. The Triangle Inequality. 2 Triangles & Inequalities. Indirect Proof. $100. $100. $100. $100. $100. $200. $200. $200. $200. $200. $300. $300. $300. $300. $300. $400. $400. $400. $400. $400. $500.

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Bisectors, Medians, and Altitudes

Inequalities and Triangles

The Triangle Inequality

2 Triangles & Inequalities

Indirect Proof

$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

answer
Answer

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

Back

bisectors medians and altitudes for 200
Bisectors, Medians, and Altitudes for $200

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

answer1
Answer

On the triangle.

Back

bisectors medians and altitudes for 300
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?

answer2
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
Bisectors, Medians, and Altitudes for $400

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

answer3
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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bisectors medians and altitudes for 500
Bisectors, Medians, and Altitudes for $500

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

answer4
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
Inequalities and Triangles for $100

Define: Comparison Property

answer5
Answer

For all real numbers a, b:

a<b, a=b, or a>b

Back

answer6
Answer

For any real numbers a and b, a>b iff there is a positive number c such that a = b + c

Back

inequalities and triangles for 300
Inequalities and Triangles for $300

If in triangle ABC, AB = 10,

BC = 12 and CA = 9, which angle has the greatest measure?

answer7
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
Inequalities and Triangles for $400

If in triangle ABC, <A = 10 degrees, <B = 85 degrees and <C = 85 degrees, which side is the longest?

answer8
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
Inequalities and Triangles for $500

Define the exterior angle inequality theorem

answer9
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
Indirect Proof for $100

Define: Indirect Reasoning

answer10
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
Indirect Proof for $200

List the three steps for writing an indirect proof:

answer11
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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indirect proof for 300
Indirect Proof for $300

Prove that there is no greatest even integer.

answer12
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
Indirect Proof for $400

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

answer13
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
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.

answer14
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
The Triangle Inequalityfor $100

Write the triangle inequality theorem:

answer15
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 Triangle Inequalityfor $200

The shortest segment from a point to a line is_______

answer16
Answer

The segement perpendicular to the line that passes through the point.

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the triangle inequality for 300
The Triangle Inequalityfor $300

Can the following lengths be sides of a triangle?

4, 5, 9

answer17
Answer

No, 4+5 = 9, in order to be a triangle 4+5 > 9

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the triangle inequality for 400
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:

answer18
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
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

answer19
Answer

Back

2 triangles inequalities for 100
2 Triangles & Inequalitiesfor $100

Write out the SAS Inequality theorem

answer20
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
2 Triangles & Inequalitiesfor $200

Write out the SSS Inequality theorem

answer21
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
2 Triangles & Inequalitiesfor $300

Given: ST = PQ, SR = QR and ST = 2/3 SP

Prove: m<SRP > m<PRQ

Q

R

T

P

S

answer22
Answer

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2 triangles inequalities for 400
2 Triangles & Inequalitiesfor $400

Given: KL || JH; JK = HL;

m<JKH + m<HKL < m<JHK + m<KHL

Prove: JH < KL

K

J

H

L

answer23
Answer

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2 triangles inequalities for 500
2 Triangles & Inequalitiesfor $500

Given: PQ is congruent to SQ

Prove: PR > SR

S

P

T

R

Q

answer24
Answer

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