Chapter 5 Properties of Triangles. Chapter 5 Objectives Identify a perpendicular bisectorIdentify a perpendicular bisector Identify characteristics of

Chapter 5Chapter 5

Properties of TrianglesProperties of Triangles

Chapter 5 ObjectivesChapter 5 Objectives

• Identify a perpendicular bisectorIdentify a perpendicular bisector

• Identify characteristics of angle bisectorsIdentify characteristics of angle bisectors

• Visualize concurrency points of a triangleVisualize concurrency points of a triangle

• Compare measurements of a triangleCompare measurements of a triangle

• Display the midsegment of a triangleDisplay the midsegment of a triangle

• Utilize the triangle inequality theoremUtilize the triangle inequality theorem

• Create an indirect proofCreate an indirect proof

Lesson 5.1Lesson 5.1

PerpendicularsPerpendiculars

andand

BisectorsBisectors

Lesson 5.1 ObjectivesLesson 5.1 Objectives

• Define perpendicular bisectorDefine perpendicular bisector

• Utilize the Perpendicular Bisector Utilize the Perpendicular Bisector Theorem and its converseTheorem and its converse

Perpendicular BisectorPerpendicular Bisector

• A segment, ray, line, or plane that is A segment, ray, line, or plane that is perpendicularperpendicular to a to a segment at its segment at its midpointmidpoint is called the is called the perpendicular perpendicular bisectorbisector..

EquidistantEquidistant

• In order for an object to be In order for an object to be equidistantequidistant from from two or moretwo or more locations, the following must be locations, the following must be true:true:

1.1. The distance to each object must be equal.The distance to each object must be equal.

2.2. The segment drawn from the object must intersect The segment drawn from the object must intersect each location at the same angle.each location at the same angle.

Theorem 5.1:Theorem 5.1:Perpendicular Bisector TheoremPerpendicular Bisector Theorem

• If a point is on the If a point is on the perpendicular bisectorperpendicular bisector of a segment, of a segment, then it is then it is equidistantequidistant from the from the endpointsendpoints of the of the segment.segment.

Theorem 5.2:Theorem 5.2:Perpendicular Bisector ConversePerpendicular Bisector Converse

• If a point is If a point is equidistantequidistant from the from the endpointsendpoints of a segment, of a segment, then it is on the then it is on the perpendicular bisectorperpendicular bisector of the segment. of the segment.

Example 1Example 1

Tell whether there is enough information that C lies Tell whether there is enough information that C lies on the on the perpendicular bisectorperpendicular bisector of segment AB. of segment AB.Explain.Explain.

Yup!Yup!

C is C is equidistantequidistant from A and B from A and B

Yup!Yup!

C is C is equidistantequidistant from A and B from A and B

Theorem 5.3:Theorem 5.3:Angle Bisector TheoremAngle Bisector Theorem

• If a point is on the If a point is on the bisector of an anglebisector of an angle, then it is , then it is equidistantequidistant from the from the two sidestwo sides of the angle. of the angle.

Theorem 5.4:Theorem 5.4:Angle Bisector ConverseAngle Bisector Converse

• If a point is on the interior of an angle and it is If a point is on the interior of an angle and it is equidistantequidistant from the from the two sidestwo sides of the angle, then it lies of the angle, then it lies on the on the bisector of an anglebisector of an angle..

Example 2Example 2

Can you conclude that ray BD bisects Can you conclude that ray BD bisects ABC?ABC?Explain.Explain.

Yup!Yup!

D is D is equidistantequidistant from A and C from A and C

Nope!Nope!

We do not know the anglesWe do not know the anglesat which the segmentsat which the segmentsintersect the sides of intersect the sides of ABC.ABC.

Lesson 5.1 HomeworkLesson 5.1 Homework

• In ClassIn Class– 1-71-7

• p268-271p268-271

• HWHW– 8-13, 16-26, 41-528-13, 16-26, 41-52

• Due TomorrowDue Tomorrow


BisectorsBisectors

of aof a

TriangleTriangle


• Define concurrencyDefine concurrency• Identify the concurrent points inside triangles.Identify the concurrent points inside triangles.• Identify perpendicular and angle bisectors in a Identify perpendicular and angle bisectors in a

triangle.triangle.• Differentiate between circumcenter and incenterDifferentiate between circumcenter and incenter

Perpendicular Bisectors of a TrianglePerpendicular Bisectors of a Triangle

• A A perpendicular bisector of a triangleperpendicular bisector of a triangle is any is any segment or ray or line that is segment or ray or line that is perpendicularperpendicular to to the midpoint of any side of a triangle.the midpoint of any side of a triangle.

ConcurrencyConcurrency

• Concurrent linesConcurrent lines exist when exist when 33 or moreor more lines, segments, or lines, segments, or rays intersect at a rays intersect at a common point.common point.

• The point at which the The point at which the concurrent linesconcurrent lines intersect is called the intersect is called the point of point of concurrencyconcurrency..

Theorem 5.5:Theorem 5.5:Perpendicular Bisectors of a TrianglePerpendicular Bisectors of a Triangle

• The The perpendicular bisectorsperpendicular bisectors of a triangle will of a triangle will intersect to form a intersect to form a point of concurrencypoint of concurrency equidistantequidistant from the from the verticesvertices..

Hint:Hint: If the segment is If the segment is perpendicularperpendicular to a to a side, then it isside, then it is equidistantequidistant to the to the vertices.vertices.

CircumcenterCircumcenter• The The point of concurrencypoint of concurrency of of perpendicular bisectorsperpendicular bisectors in a in a

triangle is called the triangle is called the circumcenter of a trianglecircumcenter of a triangle..– It is called this because it forms the It is called this because it forms the centercenter of a circle that is of a circle that is

drawn connecting the vertices of the triangle.drawn connecting the vertices of the triangle.– Notice the vertices of the triangle lie on the Notice the vertices of the triangle lie on the circumferencecircumference of the of the

circle.circle.• Thus the name Thus the name circum-centercircum-center..

Example 3Example 3

Find the following quantities:Find the following quantities:

a)a) MOMOa)a) 26.826.8

b)b) PRPRb)b) 2626

c)c) MNMNc)c) 4040

d)d) SPSPd)d) 2222

e)e) MPMPe)e) 4444

Inside OutInside Out

Type of Type of TriangleTriangle

Acute TriangleAcute Triangle Right TriangleRight Triangle Obtuse TriangleObtuse Triangle

PicturePicture

Point of Point of ConcurrencyConcurrency InsideInside On One SideOn One Side OutsideOutside

Note: All lines drawn Note: All lines drawn must must be be perpendicular bisectorsperpendicular bisectors of the triangle sides. of the triangle sides.

Theorem 5.6:Theorem 5.6:Angle Bisectors of a TriangleAngle Bisectors of a Triangle

• The The angle bisectorsangle bisectors of a triangle intersect of a triangle intersect at a point that is at a point that is equidistantequidistant from the from the sidessides of the triangle. of the triangle.

Hint:Hint: When When anglesangles are are equal, then the equal, then the distancedistance to the side to the side is equal.is equal.

Hint:Hint: But the But the perpendicular perpendicular segments are segments are notnot bisectors.bisectors.

IncenterIncenter

• The The point of concurrencypoint of concurrency of the of the angle bisectorsangle bisectors is called is called the the incenter of the triangleincenter of the triangle..– It is called this because it creates the It is called this because it creates the centercenter of a circle formed by of a circle formed by

touching each touching each sideside of the triangle of the triangle onceonce..– Notice the circle formed is Notice the circle formed is insideinside the triangle. the triangle.

• Thus the name Thus the name in-centerin-center..

Example 4Example 4

Point T is the Point T is the incenterincenter of of PQR.PQR.

Find ST.Find ST.

1515

Example 5Example 5

• Three snack carts sell Three snack carts sell frozen yogurt at locations A, frozen yogurt at locations A, B, and C. The distributor for B, and C. The distributor for the snack carts wants to the snack carts wants to build a warehouse that is build a warehouse that is equal distance to all three equal distance to all three carts.carts.

a)a) Describe how the distributor Describe how the distributor could find a location for the could find a location for the warehouse.warehouse.

b)b) Show where the warehouse Show where the warehouse should be built on the map. should be built on the map.

Use the perpendicular bisectors of a Use the perpendicular bisectors of a triangle to determine the circumcenter of triangle to determine the circumcenter of the three locations. The circumcenter is the three locations. The circumcenter is equidistant from all vertices of a triangle.equidistant from all vertices of a triangle.

Mr. Lent’s Ice Cream Warehouse



• p275-278p275-278

• HWHW– 5-21, 24-285-21, 24-28



Medians and AltitudesMedians and Altitudes

of Trianglesof Triangles


• Define a median of a triangleDefine a median of a triangle

• Identify a centroid of a triangleIdentify a centroid of a triangle

• Define the altitude of a triangleDefine the altitude of a triangle

• Identify the orthocenter of a triangleIdentify the orthocenter of a triangle

Triangle MediansTriangle Medians

• A A median of a trianglemedian of a triangle is a segment that does is a segment that does the following:the following:– Contains one endpoint at a Contains one endpoint at a vertexvertex of the triangle, and of the triangle, and– Contains the other endpoint at the Contains the other endpoint at the midpointmidpoint of the of the

opposite sideopposite side of the triangle. of the triangle.A

B

CD

CentroidCentroid

• When all three medians are drawn in, they When all three medians are drawn in, they intersect to form the intersect to form the centroid of a trianglecentroid of a triangle..– This special This special point of concurrencypoint of concurrency is the balance point is the balance point

for any evenly distributed triangle.for any evenly distributed triangle.• In Physics, we would call it theIn Physics, we would call it the center of masscenter of mass..

AcuteAcute RightRight

ObtuseObtuse

Remember: All Remember: All mediansmedians intersect the intersect the midpointmidpoint of the of the opposite side.opposite side.

Theorem 5.7:Theorem 5.7:Concurrency of Medians of a TriangleConcurrency of Medians of a Triangle

• The The medians of a trianglemedians of a triangle intersect at a point intersect at a point that is two-thirds of the distance from each that is two-thirds of the distance from each vertex to the midpoint of the opposite side.vertex to the midpoint of the opposite side.– The The centroidcentroid is is 22//33 the distance from any vertex to the the distance from any vertex to the

opposite side.opposite side.

AP = AP = 22//33AEAE

BP = BP = 22//33BFBF

CP = CP = 22//33CDCD 22 // 33AEAE

22 // 33B

FB

F

22//33 CDCD

Example 6Example 6SS is the is the centroidcentroid of of RTW, RS = 4, VW = 6, and TV = 9. Find the following:RTW, RS = 4, VW = 6, and TV = 9. Find the following:a)a) RVRV

a)a) 66

b)b) RURUb)b) 66

• 4 is 4 is 22//33 of 6 of 6• Divide 4 by 2 and then muliply by 3. Divide 4 by 2 and then muliply by 3. Works everytime!!Works everytime!!

c)c) SUSUc)c) 22

d)d) RWRWd)d) 1212

e)e) TSTSe)e) 66

• 6 is 6 is 22//33 of 9 of 9

f)f) SVSVf)f) 33

AltitudesAltitudes

• An An altitude of a trianglealtitude of a triangle is the is the perpendicularperpendicular segment segment from a from a vertexvertex to the to the opposite sideopposite side..– It It does not does not bisectbisect the the angleangle..– It It does notdoes not bisect the bisect the sideside..

• The The altitudealtitude is often thought of as the is often thought of as the heightheight..• While true, there are While true, there are 33 altitudesaltitudes in every triangle but only in every triangle but only

11 heightheight!!

OrthocenterOrthocenter

• The three The three altitudesaltitudes of a triangleof a triangle intersect at a point that intersect at a point that we call the we call the orthocenter of the triangleorthocenter of the triangle..

• The The orthocenterorthocenter can be located: can be located:– inside the triangleinside the triangle– outside the triangle, oroutside the triangle, or– on one side of the triangleon one side of the triangle

AcuteAcute

RightRightObtuseObtuse

The The orthocenterorthocenter of a right of a right triangle will always be located at triangle will always be located at the vertex that forms the right the vertex that forms the right angle.angle.

Theorem 5.8:Theorem 5.8:Concurrency of Altitudes of a TriangleConcurrency of Altitudes of a Triangle

• The lines containing the The lines containing the altitudes of a altitudes of a triangletriangle are are concurrentconcurrent..

Example 7Example 7

Is segment BD a median, altitude, or perpendicular bisector of Is segment BD a median, altitude, or perpendicular bisector of ABC?ABC?Hint: It could be more than one!Hint: It could be more than one!

MedianMedian

AltitudeAltitude

PerpendicularPerpendicularBisectorBisector

NoneNone



• p282-284p282-284

• HWHW– 8-23, 39-458-23, 39-45


• Quiz TuesdayQuiz Tuesday– November 20November 20


Midsegment TheoremMidsegment Theorem

ofof

TrianglesTriangles


• Create the midsegment of a triangleCreate the midsegment of a triangle

• Identify the characteristics of a Identify the characteristics of a midsegment of a trianglemidsegment of a triangle

Midsegment of a TriangleMidsegment of a Triangle

• So far we have studied So far we have studied 44 types of special segments of triangles. types of special segments of triangles.1.1. Perpendicular BisectorPerpendicular Bisector

2.2. Angle BisectorAngle Bisector

3.3. MedianMedian

4.4. AltitudeAltitude– It just so happens that all of these intersect only one side at a time.It just so happens that all of these intersect only one side at a time.

– And three of the four intersect an vertex and a side.And three of the four intersect an vertex and a side.

• Another type of special segment is one that connects the Another type of special segment is one that connects the midpoints of the sides of a triangle.midpoints of the sides of a triangle.

– This special segment is called the This special segment is called the midsegment of a trianglemidsegment of a triangle..• Notice there are Notice there are 33 midsegmentsmidsegments in every triangle. in every triangle.

Theorem 5.9:Theorem 5.9:Midsegment Theorem of a TriangleMidsegment Theorem of a Triangle

• The segment connecting the midpoints of the two sides The segment connecting the midpoints of the two sides of a triangle is:of a triangle is:– ParallelParallel to the to the third sidethird side– HalfHalf the the lengthlength of the of the third sidethird side

• The side it is The side it is parallelparallel to to

Segment DE // Segment ACSegment DE // Segment AC

DE = DE = 11//22ACAC

Example 8Example 8

Segment MP is the midsegment of Segment MP is the midsegment of LNO.LNO.

Find xFind x

MP = ½NO MP = ½NO

x = ½(16) x = ½(16)

x = 8 x = 8

MP = ½NO MP = ½NO

7 = ½(x) 7 = ½(x)

x = 14 x = 14

P is the midpointP is the midpoint

x = 4x = 4

Example 9Example 9Fill in the followingFill in the followinga)a) Segment GJ is parallel to ________.Segment GJ is parallel to ________.

a)a) segment DFsegment DF

b)b) Segment EJ is congruent to _________.Segment EJ is congruent to _________.b)b) segment JFsegment JF

c)c) Segment DE is parallel to __________.Segment DE is parallel to __________.c)c) segment KJsegment KJ

d)d) If EF = 18, then GK = _____.If EF = 18, then GK = _____.d)d) 99

e)e) If JK = 13, then ED = _____.If JK = 13, then ED = _____.e)e) 2626



• p290-293p290-293

• HWHW– 12-18, 21-29, 39-49 odds12-18, 21-29, 39-49 odds



InequalitiesInequalities

inin

One TriangleOne Triangle


• Compare angle sizes based on side Compare angle sizes based on side lengthslengths

• Utilize the Triangle Inequality TheoremUtilize the Triangle Inequality Theorem

Theorem 5.10:Theorem 5.10:Side Lengths of a Triangle TheoremSide Lengths of a Triangle Theorem

• If one side of a triangle is longer than another If one side of a triangle is longer than another side, then the angle opposite the longer side is side, then the angle opposite the longer side is larger than the angle opposite the shorter side.larger than the angle opposite the shorter side.– Basically, the larger the side, the larger the angle Basically, the larger the side, the larger the angle

opposite that side.opposite that side.

Longest side

Longest side

Largest AngleLargest Angle 22ndnd Longest Side Longest Side

22ndnd Largest Largest AngleAngle

Smallest Smallest SideSide

Smallest Smallest AngleAngle

Theorem 5.11:Theorem 5.11:Angle Measures of a Triangle TheoremAngle Measures of a Triangle Theorem• If one angle of a triangle is larger than another If one angle of a triangle is larger than another

angle, then the side opposite the larger angle is angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.longer than the side opposite the smaller angle.– Basically, the larger the angle, the larger the side Basically, the larger the angle, the larger the side

opposite that angle.opposite that angle.

Longest side

Longest side

Largest AngleLargest Angle 22ndnd Longest Side Longest Side

22ndnd Largest Largest AngleAngle

Smallest Smallest SideSide

Smallest Smallest AngleAngle

Example 10Example 10

Name the smallest and largest angle.Name the smallest and largest angle.

SmallestSmallest

LargestLargest

SmallestSmallestLargestLargest

LargestLargestSmallestSmallest


Name the smallest and largest side.Name the smallest and largest side.

Sm

allestS

mallest

LargestLargest

Largest

Largest

SmallestSmallest

Theorem 5.13:Theorem 5.13:Triangle InequalityTriangle Inequality

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

Add each combination of two sides to make Add each combination of two sides to make sure that they are longer than the third sure that they are longer than the third remaining side.remaining side.

66 66 66

113333 22 44 44


Determine whether the following could be lengths Determine whether the following could be lengths of a triangle.of a triangle.

a)a) 6, 10, 156, 10, 15a)a) 6 + 10 > 156 + 10 > 15

10 + 15 > 610 + 15 > 66 + 15 > 106 + 15 > 10YES!YES!

b)b) 11, 16, 3211, 16, 32b)b) 11 + 16 < 3211 + 16 < 32

NO!NO!Hint: A shortcut is to make sure that the Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than sum of the two smallest sides is bigger than the third side.the third side.

The other sums will always work.The other sums will always work.

Homework 5.5Homework 5.5


• p298-301p298-301

• HWHW– 6-30 evens6-30 evens

• Due End of the HourDue End of the Hour

Documents

Chapter 5 Properties of Triangles. Chapter 5 Objectives Identify a perpendicular bisectorIdentify a perpendicular bisector Identify characteristics of