Perpendiculars and Bisectors Answer A. Bisector 2. 1. of - - PDF document

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Geometry Triangles

www.njctl.org 2014-02-12

Slide 3 / 165 Triangles Table Of Contents

· Perpendiculars And Bisectors Of Segments And Angles · Perpendicular Bisectors Of A Segment · Angle Bisectors · Bisectors Of A Triangle · Medians Of A Triangle · Altitudes Of A Triangle · Points Of Concurrency · Midsegments Of A Triangle · Inequalities In One Triangle · Hinge Theorem · Indirect Proof

Click on a title to link to that page.

Slide 4 / 165 Triangles Table Of Contents Labs

· Perpendicular Bisector Theorem Lab

Click on a title to link to that page.

· Angle Bisector Theorem Lab · Perpendicular Bisectors of a Triangle Labs · Angle Bisectors of a Triangle Labs · Medians of a Triangle Labs · Altitudes of a Triangle Lab · Special Cases Lab & Sketch · Midsegments of a Triangle Labs · Inequalities in One Triangle Lab & Sketch · Triangle Inequality Lab & Sketch

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Perpendiculars and Bisectors

• f Segments and Angles

Return to Table

• f Contents

Slide 6 / 165 Vocabulary Review

Match the correct term for the red line in the sketch on the left with the terms on the right.

• A. Bisector
• B. Perpendicular
• C. Perpendicular Bisector
• D. Angle Bisector

Answer 1. 2. 3. 4.

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Term Definition Diagram

Bisector of a segment

A segment, ray, line or plane that divides a segment into 2 equal parts.

Perpendicular to a segment

A segment, ray, line or plane that intersects a segment at right angles.

Perpendicular Bisector of a segment

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

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Term Definition Diagram

Angle Bisector

A ray that divides and angle into 2 adjacent, congruent angles.

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1 A bisector of a segment is also a perpendicular bisector

• f a segment.

True False

Answer

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2 If AB is the bisector of XY then XM is 7. True False

Answer

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3 HJ is the angle bisector of AHB, find the measure of AHB. A 75 B 21 C 150 D none of these

Answer

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4 The perpendicular bisector of a segment intersects the segment at its _____. A endpoint B midpoint C top D bottom

Answer

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Perpendicular Bisectors

• f a Segment

Return to table

• f contents

Slide 14 / 165 The Perpendicular Bisector Theorem

To investigate the perpendicular bisector theorem go to the lab titled, "The Perpendicular Bisector Theorem Lab."

Go to the "Perpendicular Bisector Theorem Lab" Click here to review constructing a perpendicular bisector of a segment.

"math is fun"

Slide 15 / 165 The Perpendicular Bisector Theorem

A point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

XM is the perpendicular bisector of AB therefore XA ≅ XB

Slide 16 / 165 Fill in the blanks according to the diagram below.

CD is the _____ of AB therefore AC ≅_____ and BC = _____ ? ? ? Answer

Slide 17 / 165 Discuss Why JM ≅ LM Using Congruent Triangles.

Given: KM is the perpendicular bisector of JL. Prove: JM LM ≅

Remember when proving the perpendicular bisector theorem you may not use it as a reason in your proof.

Answer

Slide 18 / 165 If JM ML, and K is the midpoint of JL is it possible to prove M lies on the perpendicular bisector of JL?

≅

Hint: Construct segment KM.

Discuss a plan for the proof of the Converse of the Perpendicular Bisector Theorem. Answer

Slide 19 / 165 The Converse of the Perpendicular Bisector Theorem

If JM ML, then M lies on the perpendicular bisector of JL. ≅

If a point is equidistant from the endpoints of a segment then it lies

• n the perpendicular bisector of the segment.

Slide 20 / 165 Fill in the blanks according to the diagram below.

PS≅ therefore S lies on the of PR and PQ≅ ? ? ? Answer

Slide 21 / 165 Using the diagram below, find JK, KL and KM.

Answer

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Emani J ada Main Street Main Street

Karen

Central Avenue Prospect Street Park Place Harrison Street Harrison Street

Emani and Jada both live in the same town on the same street, Main Street. Emani lives on the corner of Central and Main and Jada lives on the corner of Prospect and Main. Their friend Karen lives on Park Place and Harrison Street . Karen lives the same distance from both Emani and Jada. What conclusion can you make about the relationship between Park Place and Main Street? Answer

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5 Tell whether the information in the diagram allows you to conclude that X is on the perpendicular bisector of PQ. Explain your reasoning. Yes No

Answer

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6 Tell whether the information in the diagram allows you to conclude that X is on the perpendicular bisector of PQ. Yes No

Answer

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7 In the diagram, XY AB and AY YB. Find YB and XA. A 9, 40 B 4.5, 40 C 4.5, 41 D 9, 41

≅ Answer

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8 In the diagram XY is the perpendicular bisector of AB. Because AZ=BZ=15, what can you conclude about point Z? A ZY = 15 B Z is on XY, the perpendicular bisector of AB. C ZY = 9 D No conclusion can be drawn.

Answer

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Angle Bisectors

Return to Table

• f Contents

Slide 28 / 165 The Angle Bisector Theorem

To investigate the angle bisector theorem go to the lab titled, "The Angle Bisector Theorem Lab."

Go to the "Angle Bisector Theorem Lab" Click here to review constructing an angle bisector.

"math open reference"

Slide 29 / 165 The Angle Bisector Theorem

A point on the angle bisector is equidistant from the sides

• f the angle.

BX is the angle bisector of <ABC therefore XY ≅ XZ and <XYB and <XZB are right angles.

Slide 30 / 165 Fill in the blanks according to the diagram below.

EG is the _____ of <FEH and GF is _____ to EF and GH is _____ to EH therefore FG≅ _____. ? ? ? ? Answer

Slide 31 / 165 Tell whether the information in the diagram allows you to conclude that A is

• n the bisector of Y.

Answer

Slide 32 / 165 If X lies on the interior of ABC and is equidistant from the sides of ABC does point X lie on the angle bisector of ABC?

Remember: For a point to be equidistant to a ray it must be the perpendicular distance. Hint: Construct BX

Discuss a plan for the proof of the converse of the angle bisector theorem. Answer

Slide 33 / 165 Converse of the Angle Bisector Theorem

If a point lies on the interior of an angle and is equidistant from the sides of the angle then the point lies on the angles bisector of the angle.

If X is on the interior of ABC and equidistant from the sides of ABC then X lies on the angle bisector of ABC.

Slide 34 / 165 Fill in the blanks according to the diagram below.

EH _____EF and GH ______GF also EH _____ GH therefore H is _____ from the sides of EFG and lies on the _____ of EFG. ? ? ? ? ? Answer

Slide 35 / 165 Tell whether the information in the diagram allows you to conclude that J lies on the angle bisector of KLM.

Answer

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9 Is there enough information to conclude whether or not point D is on the bisector of B? Yes No

Answer

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10 Is there enough information to conclude whether or not point D is on the bisector of B? Yes No

Answer

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11 In the diagram, AB bisects <DAF, BE AE, BG AG, and BE = 2. Find BG. A 2 B 3 C 6 D None of the above

Answer

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12 In the diagram, AB bisects DAF, CD AD, CF AF, and CD=CF=6. What can you conclude about point C? A No conclusion can be drawn. B C lies on the bisector of DAF. C C does not lie on the bisector of DAF. D C lies on the bisector of AD.

Answer

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Bisectors of a Triangle

There are Two Types:

• 1. Perpendicular Bisectors of a Triangle
• 2. Angle Bisectors of a Triangle

Return to Table

• f Contents

Slide 41 / 165 Perpendicular Bisectors of a Triangle

To investigate the perpendicular bisectors of a triangle go to the lab titled, "The Perpendicular Bisectors of a Triangle Lab."

Go to the "Concurrency of Perpendicular Bisectors of a Triangle Lab" using gsp. Go to the "Concurrency of Perpendicular Bisectors of a Triangle Lab" using paper-folding.

Slide 42 / 165 Concurrency of Perpendicular Bisectors of a Triangle Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

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The point of intersection of the lines is called the point of concurrency. Are the three perpendicular bisectors of a triangle concurrent?

click here

When three or more lines (or rays, or segments) intersect at the same point, they are called concurrent.

click here

Answer

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The point of concurrency is called the circumcenter because it also happens to be the center of the circle that is circumscribed around the triangle.

Cicumcenter

click here

Slide 45 / 165 Cicumcenter

P is the circumcenter of ABC.

P is also the center of the circle with radii PA, PB, and PC. Therefore the circumcenter (P) is equidistant from the vertices of ABC.

Slide 46 / 165 Cicumcenter

Acute Triangle ABC with circumcenter P.

Slide 47 / 165 Cicumcenter

Right Triangle ABC with circumcenter P.

Slide 48 / 165 Cicumcenter

Obtuse Triangle ABC with circumcenter P.

Slide 49 / 165 Cicumcenter

If the circumcenter is located in the interior of ABC then ABC is acute. If the circumcenter is located outside of ABC then ABC is

• btuse.

Where is the circumcenter located if ABC is right? The circumcenter is located on the midpoint of the hypotenuse.

Slide 50 / 165 Cicumcenter

ACUTE RIGHT OBTUSE

Slide 51 / 165 Find the circumenter of ABC with vertices:

A (0,0) B (12,6) C (18,0)

Hint: Find the equation of the line that contains AC and AB. Solve the system of equations to find their point of intersection which is the circumcenter of ABC.

click here

Answer

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13 If three or more lines intersect at the same point, the lines are _____. A locus B circumcenter C collinear D concurrent

Answer

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14 The perpendicular bisectors of ABC meet at point G. Find GC. A 2 B 5 C 7 D Not enough information.

A B C D E F G 5 2 7

Answer

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15 The perpendicular bisector of a triangle _____ passes through through the midpoint of a side of the triangle. A Always B Sometimes C Never

Answer

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16 The circumcenter of a triangle _____ lies outside the triangle A Always B Sometimes C Never

Answer

Slide 56 / 165 Angle Bisectors of a Triangle

To investigate the angle bisectors of a triangle go to the lab titled, "The Angle Bisectors of a Triangle Lab."

Go to the "Concurrency of Angle Bisectors

• f a Triangle Lab" using gsp.

Go to the "Concurrency of Angle Bisectors

• f a Triangle Lab" using

paper-folding.

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Are the three angle bisectors of a triangle concurrent?

Answer

Slide 58 / 165 Concurrency of Angle Bisectors of a Triangle Theorem

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

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The point of concurrency is called the incenter because it also happens to be the center of the circle that is inscribed in the triangle.

Incenter

click here

Slide 60 / 165 Incenter

P is the ___ of ABC

P is also the ____ of the of the inscribed circle with radii PX, PY, and PZ. Therefore the incenter (P) is _____ from the sides of ABC. ? ? ? Answer

Slide 61 / 165 Incenter

Acute Triangle ABC with incenter P.

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Right Triangle ABC with incenter P.

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Obtuse Triangle ABC with incenter P.

Slide 64 / 165 Incenter

True or False: The incenter of a right triangle lies on the vertex of the right angle. Answer

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17 The angle bisectors of PQR meet at point T. Find TK.

A 12 B 8 C 5 D 1 Answer

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18 In the diagram below of PQR, RA is the bisector

• f QRP, PB is the bisector of RPQ, and QT is
• drawn. Which statement must be true?

A AT=BT B PT=QT C PBQ PBR D AQT BQT Answer

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19 The incenter of a triangle is equidistant from the vertices

• f the triangle.

True False

Answer

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20 What is the point of concurrency of the perpendicular bisectors of a triangle called? A Center B Midpoint C Circumcenter D Incenter

Answer

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21 What is the name of point Q? A Perpendicular Bisector B Incenter C Center D Circumcenter

Q Answer

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Medians of a Triangle

Return to Table

• f Contents

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M D F E Median - The median connects a vertex of the triangle to the midpoint

• f the side opposite that vertex.

Vocabulary

ME is a median of DEF.

click here

Slide 72 / 165 Median

Click here to review construction of a median of a triangle.

M D F E

"math open reference"

Slide 73 / 165 Medians of a Triangle

To investigate the medians of a triangle go to the lab titled, "The Medians of a Triangle Lab."

Go to the "Concurrency of Medians

• f a Triangle Lab" using gsp.

Go to the "Concurrency of Medians

• f a Triangle Lab" using

paper-folding.

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Are the medians of a triangle concurrent?

Answer

Slide 75 / 165 Concurrency of Medians of a Triangle Theorem

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. If P is the centroid of ABC then, AP=(2/3)AF, BP = (2/3)BG and CP = (2/3)CE

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Centroid

The centroid is also known as, "the center of gravity" of a

• triangle. It is known as such

because it is the balancing point

• f any triangle.

If you cut a triangle out of a piece

• f cardboard and balance it on

your pencil, your pencil will mark the location of the triangle's centroid.

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Centroid

What is the ratio of AP to AF? Answer

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Centroid

What is the ratio of PF to AF? Answer

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Centroid

What is the ratio of PF to AP? Answer

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Centroid

The centroid is the point of concurrency of the medians

• f a triangle.

The centroid is 2/3 of the distance from the vertex to the midpoint of the opposite side.

click here click here

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Centroid

The distance from the centroid to the vertex is twice as long as from the centroid to the midpoint of the opposite side.

click here

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Centroid

Acute triangle ABC with centroid P.

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Centroid

Right triangle ABC with centroid P.

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Centroid

Obtuse triangle ABC with centroid P.

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Point Q is the centroid of triangle ABC. Find the lengths of the following.

• 1. QE = _____

?

• 2. AE = _____

?

• 3. QF = _____

?

• 4. BF = _____

?

• 5. CQ = _____

?

• 6. CD = _____

? Answer

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22 Point Q is the centroid of ABC and BQ = 6, find BF. A 6 B 9 C 12 D 15

Q

B E D F C A

Answer

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23 Point Q is the centroid of ABC and QE = 11, find QA. A 5.5 B 11 C 22 D 33

Q

B E D F C A

Answer

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24 Point Q is the centroid of ABC and CD = 8, find CQ. A 8/3 B 16/3 C 8 D 16

Q

B E D F C A

Answer

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25 The perpendicular bisectors of RST meet at point Q. Find QS. A 6 B 10 C 12 D 16

Q S R T

6 16

Answer

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26 The angle bisectors of PRQ meet at point H. Find HS. A 3 B 4 C 5 D 8 H P R Q

5 3

S T U

Answer

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Altitudes of a Triangle

Return to Table

• f Contents

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Altitude - The Altitude is a perpendicular segment that connects a vertex to the side of the triangle that is opposite that vertex .

G H J K

Vocabulary

click here

HJ is the altitude of GHK

Slide 93 / 165 Altitude

Click here to review construction of a altitude of a triangle. G H J K

"math open reference"

Slide 94 / 165 Altitudes of a Triangle

To investigate the altitudes of a triangle go to the lab titled, "The Altitudes of a Triangle Lab."

Go to the "Concurrency of Altitudes

• f a Triangle Lab" using gsp.

Slide 95 / 165 Concurrency of Altitudes of a Triangle Theorem

The lines containing the altitudes of a triangle are concurrent. The point of concurrency of the altitudes of a triangle is called the orthocenter.

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Orthocenter

Acute triangle ABC with

• rthocenter at point P

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Orthocenter

Right triangle ABC with

• rthocenter at point P

Why is the orthocenter located

• n the vertex of the right angle?

Answer

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Orthocenter

Obtuse triangle ABC with

• rthocenter at point P

Why does the orthocenter lie

• utside of the triangle in an
• btuse triangle?

Answer

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Orthocenter

The orthocenter is the point of concurrency

• f the altitudes of a triangle.

In an acute triangle the

• rthocenter lies in the interior
• f the triangle.

click here click here

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Orthocenter

In a right triangle the

• rthocenter lies on the vertex
• f the right angle

In an obtuse triangle the

• rthocenter lies outside of the

triangle.

click here click here

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27 In which triangle do the three altitudes intersect outside the triangle? A A right triangle B An acute triangle C An obtuse triangle D An equilateral triangle

Answer

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28 Which two points of concurrency could be located on the triangle? A incenter and centroid B centroid and orthocenter C incenter and circumcenter D circumcenter and orthocenter

Answer

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29 The perpendicular bisectors of EFG meet at point H. Find HG. A 5 B 12 C 13 D 24 E G F

5

12 H

Answer

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30 What is the name of point R? A midpoint B circumcenter C angle bisector D incenter

R Answer

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Points of Concurrency

Return to Table

• f Contents

Slide 106 / 165 Points of Concurrency

To investigate the points of concurrency of an equilateral and isosceles triangle go to the lab titled, "Points of concurrency lab"

Go to the investigation "Points of Concurrency - Special Cases." Go to the Sketch "Points of Concurrency - Special Cases."

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Points of Concurrency Equilateral Triangle

In equilateral triangle ABC, point P is the incenter, circumcenter, centroid and

• rthocenter.

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Points of Concurrency Isosceles Triangle

Isosceles triangle ABC, where B C O - orthocenter I - Incenter Ce - centroid Ci - circumcenter

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Points of Concurrency Isosceles Triangle

Isosceles triangle ABC, where B C The incenter, orthocenter, centroid, and circumcenter lie on the same line in an isosceles triangle.

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Points of Concurrency All Triangles

Triangle ABC with: O - orthocenter Ce - centroid Ci - circumcenter I - incenter

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Points of Concurrency All Triangles

Triangle ABC with: The orthocenter, centroid, and circumcenter are collinear in a triangle.

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31 Identify the point of concurrency (P) from the given information. A incenter B circumcenter C centroid D orthocenter

Answer

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32 Identify the point of concurrency (P) from the given information in the right triangle. A incenter B circumcenter C centroid D orthocenter

Answer

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33 Identify the point of concurrency that is equidistant from the sides of the triangle. A incenter B circumcenter C centroid D orthocenter

Answer

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Midsegments of a Triangle

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Midsegment -

A Midsegment is a segment that connects the sides of a triangle at their midpoints

Vocabulary

Slide 117 / 165 Midsegments of a Triangle

To investigate the altitudes of a triangle go to the lab titled, "The Altitudes of a Triangle Lab."

Go to the "Midsegments of a Triangle Lab" using gsp. Go to the "Midsegments of a Triangle Lab" using paper-folding.

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Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. DE || AB and DE = 1/2 AB

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J K L Y Z X

XZ, YZ, and YX are midsegments of ΔJKL. Match the sides that are parallel to each other.

JK || _________ YZ YX XZ LK || _________ JL || _________

Midsegments

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Midsegments

DE, EF, and DF are midsegments

• f ABC.

Find length of:

• 1. EF = ______
• 2. BF = ______
• 3. DE = _____
• 4. BC = _____
• 5. BE = _____

Answer

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R P N Q L M

The midpoints of the sides of ΔPQR are the points L, M and N. Fill in the missing answers.

If RP = 32, then MN = _______ If LN = 7.5, then RQ = _______ If QN = 17, then LM = _______ 32 3.75 34 11 30 8 20 2.75 16 7.5 15 8.5 5 12 10 3 17 5.5 6 64

Midsegments

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R P N Q L M

The midpoints of the sides of ΔPQR are the points L, M and N. Fill in the missing answers.

If PN = 10, then PQ = _______ If RM = 5.5, then MQ = _______ If PL = 6, then RL = _______ 32 3.75 34 11 30 8 20 2.75 16 7.5 15 8.5 5 12 10 3 17 5.5 6 64

Midsegments

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R P N Q L M

L, M and N are the midpoints

• f the sides of ΔPQR.

If MN = 2X + 4 and RP = 6X - 2, then MN = _____

Midsegments

? Answer

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Midsegments

How does the triangle formed by the midsegments of triangle ABC form 4 congruent triangles? Answer

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34 L, M, and N are midpoints of the sides of PQR. If LN = x - 1 and RQ = 3x - 13, find the length of LN. A 22 B 20 C 11 D 10

R P N Q L M

Answer

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35 P, Q, and R are midpoints of the sides of ABC. If PB = 9, then RQ = ? A 18 B 36 C 4.5 D 9

C A B Q P R

Answer

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36 Given QR=16, ST=10 and SQ=7. Find the perimeter of PQR. A 25 B 50 C 33 D 66

P Q R U T S Answer

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Sierpinski Triangle

The Sierpinski Triangle is a fractal design of a triangle that is made by constructing midsegments to make new triangles, and then constructing midsegments of each of those triangles and the pattern is repeated until we have smaller and smaller fragmented triangles.

Fractal

is a geometric shape that can be made into smaller fragmented parts that look identical to its original figure Click to See the Interactive Sierpinski Triangle

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Click to See the Interactive Koch Snowflake

Koch Snowflake

The Koch Snowflake is made by removing the inner third section from each side of an equilateral triangle, and then forming new

• triangles. The pattern is continuously repeated until a figure that

resembles a snowflake is formed.

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Inequalities in one Triangle

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Slide 131 / 165 Inequalities in one Triangle

To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in one triangle"

Go to the sketch, "Inequalities in one triangle." Go to the worksheet, "Inequalities in one triangle."

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Inequalities in One Triangle Theorem

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle

• pposite the shorter side.

m A > m C Because BC > AB

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Inequalities in One Triangle Theorem

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side

• pposite the smaller angle.

EF > DE Because m D > m F

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Inequalities in One Triangle

Write the measurements of the triangles in order from least to greatest. Answer

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Inequalities in One Triangle

Write the measurements of the triangles in order from least to greatest. Answer

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Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. m 1 > m 3 and m 1 > m 4

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Exterior Angle Inequality

Why is m<1 > m<3 and m<1 > m<4? m 1 + m 2 = 180 m 2 + m 3 + m 4 = 180 m 1 + m 2 = m 2 + m 3 + m 4 m 1 = m 3 + m 4 If m 1 is equal to the sum of m 3 and m 4 then it is greater than either one.

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Exterior Angle Inequality

• 1. Write an equation about the

angle measures labeled in the diagram.

• 2. Write two inequalities about

the angle measures labeled in the diagram. Answer

Slide 139 / 165 Triangle Inequality

To investigate triangle inequality download the sketch, "triangle inequality" and the worksheet, "triangle inequality."

Go to the sketch, "Triangle Inequality." Go to the worksheet, "Triangle Inequality"

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Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC BC + AC > AB AC + AB > BC

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Triangle Inequality

Is it possible for a triangle to have side lengths of 15, 22, and 30? Answer

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Triangle Inequality

The side lengths of a triangle are 7, 15 and x. What is the range of possible values for x? Answer

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37 List the sides in order from shortest to longest. A AB < BC < AC B BC < AC < AB C AC < BC < AB D AC < AB < BC

Answer

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38 List the angles from smallest to largest. A m A < m B < m C B m C < m B < m A C m B < m C < m A D m A < m C < m B

Answer

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39 List the sides in order from shortest to longest. A a, b, c, d, e B d, e, c, b, a C d, e, a, b, c D a, b, c, e, d

Answer

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40 List the sides in order from shortest to longest. A a, b, c, d, e B e, d, c, b, a C e, d, c, a, b D b, a, c, d, e

Answer

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41 Find possible measures for XY in XYZ if XZ = 16 and YZ = 16. A 16 B 0 < XY < 32 C 0 < XY < 16 D 16 < XY < 32

Answer

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Hinge Theorem

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• f Contents

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Vocabulary

Included Angle - The angle located between two specific sides in a triangle.

K M L

K is the included angle between MK and KL M is the included angle between LM and MK L is the included angle between KL and LM

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6 14

H J K

6 14

L N M When two sides of one triangle ( ΔLMN) are congruent to two sides

• f another triangle (

ΔHJK), and the measure of the included angle

• f the first triangle is greater than the measure of the included

angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. LN ≅HK NM ≅ KJ m N > m K Therefore, LM > HJ

Hinge Theorem

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6 14

H J K

6 14

L N M When two sides of one triangle ( ΔLMN) are congruent to two sides

• f another triangle (

ΔHJK), and the third side of the first triangle is longer than the third side of the second, then the included angle

• f the first is larger than the included angle of the second.

LN = HK NM = KJ LM > HJ Therefore, m N > m K

Converse of Hinge Theorem

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6 14

H J K

95o S h

• r

t e r s i d e

6 14

L N M

135o L

• n

g e r s i d e

LN = HK NM = KJ m N > m K Therefore, LM > HJ Question : Which side is longer LM or HJ? (use the Hinge Theorem)

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Question : Which angle has the larger measure m<Q or m<T ? (use the Converse of the Hinge Theorem)

19 23

S U T

32 19 23

P Q R

40

PQ = ST QR = TU PR > SU Therefore, m Q > m T

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42 AC ____ DF A < B > C =

9

A B C

115o

14 6 14

D F E

100o

? Answer

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43 AB ____ PQ A < B > C =

112

• 95o

P Q A B ? Answer

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44 m 1 ____ m 2 A > B < C =

?

1 21 2 13

Answer

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Indirect Proof (Proof by Contradiction)

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• f Contents

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Proof

· We try to prove a statement is true by using information that will support the statement

Indirect Proof

· We try to prove a statement is true by using information

that contradicts the statement.

· We prove the statement is true by contradiction. · We start by assuming the opposite is true. · If it is impossible for the opposite to be true, then we have

proved that the original statement is true.

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Mystery

How do we know when to use Indirect Proof (Proof by Contradiction) ?

Clue

· If the statement we are asked to prove has the word " not "

then an indirect proof will be the best way to prove the statement by using a contraction (assuming the opposite is true).

· But indirect proof can also be used if we don't have the word

"not" in the statement, because we will assume the opposite is true.

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• 1. Assume ABC has more than one obtuse angle,

m A > 90 and m B > 90

• 2. m A + m B > 180

Example of an indirect proof. Given: ABC Prove: ABC does not have more than one obtuse angle. Teacher Notes

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• 3. You know that m A + m B +m C = 180 by Triangle Sum

Theorem so we cannot have more than one obtuse angle in a triangle. Example of an indirect proof. Given: ABC Prove: ABC does not have more than one obtuse angle.

• 4. Therefore we a have proved by contradiction that ABC does

not have more than one obtuse angle. Teacher Notes

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Indirect Proof

What would be your first statement in an indirect proof where you are trying to prove m || n? Answer

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Indirect Proof

What would be your first statement in an indirect proof where you are trying to prove ABC is not isosceles? Answer

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45 What is the first statement in the following indirect proof? Given 1 and 2 are supplementary. Prove: m || n A Assume 1 and 2 are not supplementary B Assume 1 and 2 are supplementary C Assume m || n D Assume m is not parallel to n

Answer

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Statements Reasons

• 1. AB is not BC and AD || BC
• 1. Given
• 3. 2 3
• 3. ______
• 4. 1 3
• 4. Transitive property
• 5. AB BC
• 5. ______
• 6. 1 is not 2
• 6. contradiction to the given

Finish the following indirect proof. Given: AB is not BC and AD || BC Prove: 1 is not 2

• 2. Assume _____
• 2. Assumption leading to contradiction

Answer