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Bisectors, Medians, and Altitudes

Inequalities and Triangles

2 Triangles & Inequalities

Indirect Proof The Triangle Inequality

Bisectors, Medians, and Altitudesfor $100

Define: orthocenter

Answer

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

Back

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

Back

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

Bisectors, Medians, and Altitudes for $500

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

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) = 8GD = (1/3)*(AD) = (1/3)(12) = 4

Answer

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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

Back

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

Back

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.

Inequalities and Triangles for $500

Define the exterior angle inequality theorem

Answer

Back

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

Indirect Proof for $100

Define: Indirect Reasoning

Answer

Back

Indirect reasoning – reasoning that assumes the conclusion is false and then shows that this assumption leads to a contradiction.

Indirect Proof for $200

List the three steps for writing an indirect proof:

Answer

Back

List the three steps for writing an indirect proof:1) Assume that the conclusion is false2) 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

Indirect Proof for $300

Prove that there is no greatest even integer.

AnswerAssume 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 Back

Indirect Proof for $400

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

AnswerAssume 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

Back

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.

AnswerAssume 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 Inequalityfor $100

Write the triangle inequality theorem:

Answer

Back

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

The Triangle Inequalityfor $200

The shortest segment from a point to a line is_______

Answer

Back

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

The Triangle Inequalityfor $300

Can the following lengths be sides of a triangle?

4, 5, 9

Answer

Back

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

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:

Answer

Back

43 – 37 = 6

43 + 37 = 80

So the range for the third side, x, is:

6 < x < 80

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:

A

P

Bl31 2

Answer

Back

Statements ReasonsPA ┴ 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

2 Triangles & Inequalitiesfor $100

Write out the SAS Inequality theorem

Answer

Back

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.

2 Triangles & Inequalitiesfor $200

Write out the SSS Inequality theorem

Answer

Back

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.

2 Triangles & Inequalitiesfor $300

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

Prove: m<SRP > m<PRQ

T

RQ

PS

Answer

Back

Statements Reasons

SR = QR

ST = PQ

ST = 2/3 SP; SP > ST

Given

PR = PR Reflexive

SP > PQ Substitution

m<SRP > m < PRQ SSS Inequality

2 Triangles & Inequalitiesfor $400

Given: KL || JH; JK = HL;

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

Prove: JH < KLK

L

J

H

Answer

Back

Statements Reasons

m<JKH + m<HKL < m<JHK + m<KHLJK = HLKL || JH

Given

m<HKL = m < JHK Alt. Interior Angle Theorem

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

Substitution

m<JKH < m< KHL Subtraction

HK = HK Reflexive

JH < KL SAS Inequality Theorem

2 Triangles & Inequalitiesfor $500

Given: PQ is congruent to SQProve: PR > SR

Q R

PT

S

Answer

Back

Statements Reasons

PQ is congruent to SQ Given

QR = QR Reflexive Property

m<PQR = m<PQS + m<SQR Angle Addition Postulate

m<PQR > m< SQR Definition of Inequality

PR > SR SAS Inequality Theorem