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# Welcome To - PowerPoint PPT Presentation

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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Inequalities and Triangles

The Triangle Inequality

2 Triangles & Inequalities

Indirect Proof

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Define: orthocenter

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

Back

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

On the triangle.

Back

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

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

Back

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

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

Back

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) = 8

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

Back

Inequalities and Triangles for \$100

Define: Comparison Property

For all real numbers a, b:

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

Back

Inequalities and Triangles for \$200

Define: Inequality

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?

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

Back

Inequalities and Triangles for \$400

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

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.

Back

Inequalities and Triangles for \$500

Define the exterior angle inequality theorem

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

Back

Define: Indirect Reasoning

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

Back

List the three steps for writing an indirect proof:

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

Back

Prove that there is no greatest even integer.

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

Back

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

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

Back

Given: Bobby and Kina together hit at least 30 home runs. Bobby hit 18 home runs.

Prove: Kina hit at least 12 home runs.

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.

Back

The Triangle Inequalityfor \$100

Write the triangle inequality theorem:

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

Back

The Triangle Inequalityfor \$200

The shortest segment from a point to a line is_______

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

Back

The Triangle Inequalityfor \$300

Can the following lengths be sides of a triangle?

4, 5, 9

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

Back

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:

43 – 37 = 6

43 + 37 = 80

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

6 < x < 80

Back

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

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2 Triangles & Inequalitiesfor \$100

Write out the SAS Inequality theorem

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.

Back

2 Triangles & Inequalitiesfor \$200

Write out the SSS Inequality theorem

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.

Back

2 Triangles & Inequalitiesfor \$300

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

Prove: m<SRP > m<PRQ

Q

R

T

P

S

Back

2 Triangles & Inequalitiesfor \$400

Given: KL || JH; JK = HL;

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

Prove: JH < KL

K

J

H

L

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2 Triangles & Inequalitiesfor \$500

Given: PQ is congruent to SQ

Prove: PR > SR

S

P

T

R

Q

Back