unit five properties of triangles. 5.1 perpendiculars and bisectors

UNIT FIVE

PROPERTIES OF TRIANGLES

5.1 Perpendiculars and Bisectors

5.1 PERPENDICULAR BISECTORS

Perpendicular Bisector: a segment, ray, line or plane that is perpendicular to a segment at its midpoint

ABoftorbiaisCP sec

A BP

C


Perpendicular Bisector Theorem: If a point is on the perpendicular bisector, then it is equidistant from the endpoints of the segments.

CDIF IS A PERPENDICULAR BISECTOR OF THEN C IS EQUIDISTANT FROM A AND B.

A D B

C

AB

CBCA


Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector.

AB

A B

C

P

D

If DA = DB, then D lies on the perpendicular bisector of D is on CP

5.1 PERPENDICULARS AND BISECTORS

EXAMPLE 1In the diagram, MN is the perpendicularBisector of ST.A. What segment

lengths in the diagram are equal?

B. Explain why Q is on MN

M

S

N

T

Q

12

12


Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

If , then DB = DC.CADmBADm

B

A

C

D


Converse of Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

If DB = DC, then CADmBADm

B

A

C

D

5.1 PERPENDICULARS AND BISECTORS EXAMPLE 2In the diagram, PM is the

bisector of

a. What is the relationship between

b. How is the distance between point M and L related to the distance between point M and N?

LPN

?NPMandLPM

P

NL

M

5.1 Perpendiculars and Bisectors Complete the Proof. Given: D is on the bisector of

Prove: DB = DC

BACACDCABDB ,

B

A

C

DSTATEMENT REASON

1.

2.

3.

4.

5.

6.

CADBAD ACDABD

ADAD ADCADB

DCDB DCDB

5.2 Bisectors of a Triangle

5.2 BISECTORS OF TRIANGLES

PERPENDICULAR BISECTOR OF A TRIANGLE: a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

Perpendicular bisector

5.2 Bisectors of Triangles

Concurrent lines: when three or more lines (or rays or segments) intersect in the same point.

Point of Concurrency: the point of intersection of the lines.


The three perpendicular bisectors of a triangle are concurrent.

The point of concurrency is called the circumcenter.

Acute triangle Right Triangle Obtuse Triangle

P

P

P


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

PA = PB = PC

B

A C

P

5.2 Bisectors of Triangles Example 1Three people need to decide on a location to hold a

monthly meeting. They will all be coming from different places in the city, and they want to make the meeting location the same distance from each person.

a. Explain why using the circumcenter as the location for the meeting would be the fairest for all.

b. Copy the triangle and locate the circumcenter. Tell what segments are congruent.

A B

C


Example 2

The perpendicular bisectors of meet

at point G. Find GC.

ABC

A

C

E

FB

DG

5

7

2


Angle Bisector of a Triangle:a bisector of an angle of a triangle.

Incenter of a Triangle: the point of concurrency of the angle bisectors.

Incenter


Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

CE

A

B

DF

PD = PE = PF

P


Example 3

The angle bisectors of meet at point M. Find MK.

XYZ

XL

Z

K

Y

J

12 5

8

5.3 Medians and Altitudes of a Triangle

5.3 Medians and Altitudes

Median of a Triangle: a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.

Centroid: the point of concurrency of the medians

ACUTE TRIANGLE

RIGHT TRIANGLE

OBUSE TRIANGLE


Concurrency of Medians of a Triangle: 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.

A

E

BD

F

CP

CECP

BFBP

ADAP

3

23

23

2

5.3 Medians and Altitudes Example 1

P is the centroid of shown below and PT = 5. Find RT and RP.

QRS

Q

R

ST

P


Example 2

Find the coordinates of the centroid of JKL

5.3 Medians and Altitudes Altitude of a triangle: the perpendicular

segment from a vertex to the opposite side or to the line that contains the opposite side.

Orthocenter: the point of concurrency of the altitudes. The orthocenter can lie inside, outside, or on the triangle.

Orthocenter


Example 3

Where is the orthocenter located in

a. If

b. If

c. If

ABCCmBmAm

45 BmAm

110Am

5.3 Medians and AltitudesExample 4: Use the diagram to match the type of special segment

with the correct segment.

Z

Y X W V

T

U

VUYVandWZUYZW

1. Median A. ZX

2. Altitude B. ZW

3. Perpendicular Bisector C. ZV

4. Angle Bisector D. TV

5.4 Midsegment Theorem


Midsegment of a Triangle: a segment that connects the midpoints of two sides of a triangle.

A

B

C

If D is the midpoint of AB and E is the midpoint of BC, then DE is a midsegment.

D E

5.4 Midsegment Theorem Example 1Show that the midsegment MN is parallel to side

JK and is half as long.

L(6, -1)

K (4, 5)

J (-2, 3)

M

N


Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

ABDEandABDE 21

A

D

C

E

B


Example 2

and are midsegments of Find UW and RT.

VWUW .RST

S

V

R

U

T

W

812


Example 3

a. What are the coordinates of Q and R?

b. Why is QR MP?

c. What is MP? What is QR?

M (0, 0) P (c, 0)

Q R

N (a, b)


Example 4

The midpoints of the sides of a triangle are A(2, 5), B(2, 2), and C(6, 5).

a. What are the coordinates of the vertices of the triangle?

b. Find the perimeter of the triangle.

5.5 Inequalities in One Triangle

5.5 Inequalities in One TriangleTriangle Theorems If one side of a triangle is longer than another

side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

A

B

C

35

CmAm


Triangle Theorems (continued) If one angle of a triangle is larger than another

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

6040

D

E

F

DEEF

5.5 Inequalities in One TriangleExample 1: Write the measurements of the

triangles in order from least to greatest.

a. b.

H

J

G35 45

100

Q

P

R

5 8

7


Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.

A

BC

BmmandAmm 11

1


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

A

BCBCACAB

ABBCAC

ACBCAB


Example 2

Given the possible triangle side lengths, which groups could form a triangle?

a. 2cm, 2cm, 5cm b. 3cm, 2cm, 5cm

c. 4cm, 2cm, 5cm


Example 3

A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side.

5.5 Indirect Proof and Inequalities in Two Triangles

5.6 Indirect Proof and Inequalities in Two Triangles Indirect Proof: A proof in which you prove

that a statement is true by first assuming that its opposite is true.

If this assumption leads to an impossibility, then you have prove that the original statement is true.

EXAMPLE 1: Given:

Prove: does not have more than one obtuse angle.

1. Assume that has more than one obtuse angle.

2. You know, however, that the sum of the measures of all three angles is 180.

ABCABC

ABC

180

9090

BmAm

BmandAm 1.

2.

CmBmAm

CmBmAm

180

180 3.

4.

3. So, you can substitute for

The last statement is not possible; angle measures in triangles cannot be negative.

So, you can conclude that the original assumption is false. That is, triangle ABC cannot have more than one obtuse angle.

Cm180

.180 BmAminBmAm

Cm

Cm

0

180180 5.

6.

5.6 Indirect Proof and Inequalities in Two Triangles Hinge Theorem: If two sides of one

triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

R

S T

V

W X

100 80

RT > VX

5.6 Indirect Proof and Inequalities in Two Triangles Converse of the Hinge Theorem: If two sides of one

triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

7 8

D

E

F A

B

C

DC

5.6 Indirect Proof and Inequalities in Two Triangles

Example 2: Complete each with <, > or =.

a. b. c.

2___1 mm

27 1

262

K

L MN

Q P

47

45

38

37

NQKL ___ FEDC ___

E D

F C

5.6 Indirect Proof and Inequalities in Two Triangles Example 3: You and a friend are flying separate planes.

You leave the airport and fly 120 miles due west. You then change direction and fly W 30 N for 70 miles. Your friend leaves the airport and flies 120 miles due east. She then changes direction and flies E 40 S for 70 miles. Each of you has flown 190 miles, but which plane is farther from the airport?

unit five properties of triangles. 5.1 perpendiculars and bisectors

Documents

angle bisectors

perpendicular bisector

point circumcenter

bisectors of trianglesexample

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bisector ofprove

point of concurrency

perpendicular bisector5