TheoremIfThen If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the distance. D

5.1 MIDSEGMENTS OF TRIANGLES

THEOREM 5-1: TRIANGLE MIDSEGMENT THEOREM

Theorem If Then

If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the distance.

D is the midpoint of CA andE is the midpoint of CB

1

2

DE AB and

DE AB

A B

C

D E

//

A B

C

D E

2x

x

EXAMPLE 1 Find all missing side lengths, given AB =

10, BE = 7, and DE = 4

AD =

DB =

EF =

AF =

FC =

AC =

EC =

DF =

BC =

B

A

C

D

E

F

10

7

4

5

5

5

4

4

8

7

7

14

EXAMPLE 2 Solve for x if the perimeter of triangle ABC

is 42

B

A

C

D

E

F

12

3x – 3

9

12 9 (3 3) (3 3) 42x x

3x - 3

3x - 3

15 6 42x

6 27x

4.5x

HOMEWORK P. 288 #’s 10-18 even, 19-27, 29-32, 37,

40-44, 49-52

5.2 PERPENDICULAR

AND ANGLE BISECTORS

THEOREM 5-2: PERPENDICULAR BISECTOR THEOREM

Theorem If Then

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

PM AB and

MA MB

45555555555555 5PA PB

A BM

P

A BM

P

THEOREM 5-3: CONVERSE OF THE PERPENDICULAR BISECTOR THEOREM

Theorem If Then

If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.

PM AB and

MA MB

45555555555555 5PA PB

A B

P

A B

P

M

EXAMPLE 1 Calculate AC

A

B CD

6x – 104x

4 6 10x x

2 10x

5x

5x

6(5) 10AC

30 10AC 20AC

Complete Got it 1, p. 293

Ans: QR = 8

DISTANCE The distance from a point to a line is the

length of the perpendicular segment from the point to the line.

B

A

l

THEOREM 5-4: ANGLE BISECTOR THEOREM

Theorem If Then

If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

bisects ,

,

QS PQR

SP QP and

SR QR

55555555555555

55555555555555

55555555555555

SP SR

Q

P

S

R

Q

P

S

R

THEOREM 5-5: CONVERSE OF THE ANGLE BISECTOR THEOREM

Theorem If Then

If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

,

,

SP QP

SR QR and

SP SR

55555555555555

55555555555555 bisects QS PQR

55555555555555

Q

P

S

R

Q

P

S

R

EXAMPLE 2 Calculate RM

7 2 25x x


Ans: FB = 21

A

R

P

M7x

2x+25

5 25x

5x

7(5)RM

35RM

HOMEWORK P. 296 #’s 6-8, 12-22, 36, 39-46

5.3 BISECTORS IN TRIANGLES

THEOREM 5-6: CONCURRENCY OF PERPENDICULAR BISECTOR THEOREM

Theorem Diagram Symbols

Perpendicular bisectors

are concurrent at P.

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

, , PX PY and PZ

P

Z

YX

A

B C

PA PB PC

CONCURRENT, POINT OF CONCURRENCY, CIRCUMCENTER OF THE TRIANGLE, When 3 or more lines intersect at one point,

they are concurrent. The point at which they intersect is the point of concurrency.

The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

CIRCUMSCRIBED ABOUT

Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle.

P

Z

YX

A

B C

EXAMPLE 1 Find the circumcenter of the following 3

points.

Circumcenter

THEOREM 5-7: CONCURRENCY OF ANGLE BISECTOR THEOREM


Angle bisectors

are concurrent at P.

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

, , AP BP and CP

PX PY PZ X

Z

YP

A

B C

INCENTER OF A CIRCLEINSCRIBED IN The point of concurrency of the angle

bisectors of a triangle is called the incenter of the triangle.

P is the center of the circle that is inscribed in the triangle.

X

Z

YP

A

B C

EXAMPLE 2 If P is the incenter and PY = 2x-4 and PZ

= 5x-13, find BP, if PB = 3x.

X

Z

YP

A

B C

2x-4

5x-133x

2 4 5 13x x

9 3x

3 x

3(3)PB

9PB

HOMEWORK P. 305 #’s 8-12 even, 15-18, 22, 33-34,

37-40

5.4 MEDIANS AND ALTITUDES

MEDIAN OF A TRIANGLE A median of a triangle is a segment

whose endpoints are a vertex and the midpoint of the opposite side.

J

D

F

E

Median

THEOREM 5-8: CONCURRENCY OF MEDIANS THEOREM


The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.

2

32

32

3

DC DJ

EC EG

FC FH

C

H

J

G

D

F

E

CENTROID OF THE TRIANGLE In a triangle, the point of concurrency of the

medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. For any triangle, the centriod is always inside the triangle.

EXAMPLE 1 In the diagram below CE = 20, CH = 6,

and DC = 8. Find CG, FC and DJ.2

3CE GE

CG

H

J

D

F

E

220

3GE

3 2 320

2 3 2GE

30GE

30 20 10CG

206

8

10



3CH FH

CG

H

J

D

F

E

16

3FH

3 1 36

1 3 1FH

18FH

18 6 12FC

206

8

10

12



3DC DJ


Ans: a) ZC = 13.5 b) 2 : 1

CG

H

J

D

F

E

28

3DJ

3 2 38

2 3 2DJ

12DJ

206

8

10

12

ALTITUDE OF A TRIANGLE An altitude of a triangle is the

perpendicular segment from the vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.

D

FE

FE

D

FE

D

THEOREM 5-9: CONCURRENCY OF ALTITUDES THEOREM

Theorem Diagram

The lines that contain the altitudes of a triangle are concurrent.

P

F

E

D

ORTHOCENTER OF THE TRIANGLE The lines that contain the altitudes of a

triangle are concurrent at the orthocenter of the triangle. The orthocenter can be inside, on, or outside the triangle.

F

E

D

Acute Triangle

F

E

D

Right Triangle

F

E

D

Obtuse Triangle

EXAMPLE 2 Triangle ABC has vertices A (1,3), B (2,7), and C (6,3).

What are the coordinates of the orthocenter of the triangle?

Graph the points

Create two altitudes

Locate the point of intersection

The orthocenter is the point (2, 4)

Complete Got it 3, p. 311Ans: (1, 2)

REVIEW OF THE 4 CONCURRENT POINTS

HOMEWORK P. 312 #’s 8-20, 24-27, 31, 37-41, 43-44

5.6 INEQUALITIES IN

ONE TRIANGLE

PROPERTY

Comparison Property of Inequality

If and 0, then a b c c a b

b c

a

COROLLARY TO THE TRIANGLE EXTERIOR ANGLE THEOREM

Corollary If Then

The measure of an exterior angle of a triangle is greater than the measure of each of the remote interior angles

and1

1

2

3

m

m

m

m

is an exterior a1 ngle

1

2

3

THEOREM 5-10 & THEOREM 5-11

Theorem 5-10 If Then

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

m mY Z


If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

BC AC

XZ XY

X Z

Y

m mA B C

BA

EXAMPLE 1 In the diagram, list the sides in order

from smallest to largest.


Ans: OX

C

B

A

30°

110° ,AC ,BC

180 (30 110)m A 40m A

BA

THEOREM 5-12: TRIANGLE INEQUALITY THEOREM

Theorem Diagram

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

Y

X

ZXY XYZ Z YZ XXZ Y XZ YXY Z

EXAMPLE 2 Can a triangle have side lengths of 4, 4,

and 8? How about 4, 5, and 6?

Complete Got it 4, p. 327Ans: a) No b) Yes

4 4 8

8 8

No, a triangle cannot be made with side lengths of 4, 4, and 8.

4 5 6

9 6

4 6 5

10 5

6 5 4

11 4

Yes, a triangle can be made with side lengths of 4, 5, and 6. The sum of the lengths of any two sides is greater than the third.

EXAMPLE 3 If two sides of a triangle are 8 and 14, what is

the range of possible lengths for the third side?


Ans: 3 < X < 11

8 14 x

22 x

8 14x

6x

14 8x

6x

The length of a side of a triangle can never be negative so we can eliminate that answer. So the range of values is between 6 and 22.

6 22x

HOMEWORK P. 328 #’s 6-34 even, 37-39, 43, 44, 46-

49, 53-55.

5.7 INEQUALITIES IN TWO TRIANGLES

THEOREM 5-13, THE HINGE THEOREM


If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third leg is opposite the larger included angle.

m mA X

A

B

C

X Z

Y

BC YZ

THEOREM 5-14, CONVERSE OF THE HINGE THEOREM


If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.

m mA X

A

B

C

X Z

Y

BC YZ

EXAMPLE 1 What is the range of possible values for

x?


Ans: -6 < x < 24

60 5 20x

U T

S

R

60°

15

10

(5x-20)°

Find the upper limit

80 5x16 x

Find the lower limit5 20 0x

5 20x 4x

4 16x

HOMEWORK P. 336 #’s 6-9, 11-14, 16-18, 21, 24, 26-

28, 31-35, 37-39

Documents

TheoremIfThen If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half the distance. D