Use WXY, where R, S, and T are midpoints of the sides....

Ch 5 Review Game & Answers

RS || _____

If TY = 4, then RS = _____

ST || _____

If RT = 7, then XY = _____

Use ∆WXY, where R, S, and T are midpoints of the sides.

X

YW T

RS

What are TR, RS, and ST called?


RS || WY

If TY = 4, then RS = 4

ST || WX

If RT = 7, then XY = 14

Midsegments

Answers:


Thesamedistancefromtwopoints.


Answer:

Equidistant


B

C

A D

E

F

12y - 8

8y + 20

8x + 6

6x + 18

DE is the perpendicular bisector of AC. Find the indicated measures.

2x + 4y

Find AD.

Find BC.

Is B on the perpendicular bisector of AC?

State the theorem that allows you to set AD ≅ BC?


12x - 8 = 8x + 204x = 28x = 7

Therefore AD = 12(7) - 8 = 76

Answers:

Perpendicular Bisector Theorem

8x + 6 = 6x + 18


8x - 12 2x + 36

Find the value of x.

What theorem allows you to say AB ≅ BC and solve for x?

A

B

C


Answers:

8x - 12 = 2x + 366x = 48x = 8

Angle Bisectors Theorem


12

13

A

B

C

D

EF

G

x-6


What is the point of concurrency of theangle bisectors (point G)?


Answers:

Pythagorean theorem states: a2 + b

2 = c

2

122 + b

2 = 13

2

144 + b2 = 169

b2 = 25

b = 5

Therefore, FG = 5 and by the Concurrency of the Angle Bisector Theorem, FG = DG = BG

x-6 = 5x = 11

Incenter


M

N

O

P

QR

S

N, P, and R are the midpoints of ∆MOQ

Find the value of SQ.

What are QN, OR, and MP?


QN = 36

164x - 12


Answers:

SQ = QN

SQ = 36

SQ = 24

MS = MP

MS = 32

32 = 4x - 124x = 44x = 11

Medians


Thepointofintersectionofthreeormorelines,rays,orsegments.


Answer:

Point of Concurrency


7 and 24

Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.


247

x

247

x

Answers:

Case 1: x is longest side

Case 2: 24 is the longest side

By the Triangle Inequality Theorem

7 + 24 > x31 > x

By the Triangle Inequality Theorem

7 + x > 24x > 17

17 < x < 31


Q

R

S

Find the coordinates of the centroid.

Find the midpoint of SQ (call it T).

Find the length of RT.


Midpoint of SQ:

T =

=

=

Length of RT:

RT = (2-2)2 + (5-(-4))

2

= 02 + 9

2

= 81

= 9

Coordinates of centroid C:

By the Concurrency of the Medians Theorem, RC = RT

RC = 9

RC = 6

Answer:


Thepointofconcurrencyofthethreemediansofatriangle.


Answer:

Centroid


Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

4 and 11


Answer:

Case 1: x is the longest side

4 11

x

4 x

11

4 + 11 > x15 > x

Case 2: 11 is the longest side

4 + x > 11x > 7

7 < x < 15


In ∆LMN, LM = 11, MN = 18, and LN = 24. Sketch and label the triangle.

List the angles in order from smallest to largest.


L

M

N

1118

24

Smallest to largest

∠N, ∠L, ∠M

Answer:


Thepointatwhichthelinescontainingthethreealtitudes ofatriangleintersect.


Answer:

Orthocenter


X

Y

Z

T

U

V

W


22

5x-8

What is the point of concurrency of the perpendicular bisectors of a triangle?


Answers:

By the Concurrency of the Perpendicular Bisector Thm, WX = WY = WZ.

Therefore, WY = WZ5x - 8 = 225x = 30x = 6

Circumcenter


WY, 4

WX, 14

Midsegments

Equidistant

76

⊥ Bisector Thm

54

x = 8

∠ Bisector Thm

x = 11

Incenter

24

x = 12

Medians

Point of Concurrency

17 < x < 31

(2,4)

9

(2,1)

Centroid

7 < x < 15

∠N, ∠L, ∠M

Orthocenter

x = 6

Circumcenter


Need to know for the test

Definitions

MidsegmentsPerpendicular BisectorEquidistantConcurrentPoint of ConcurrencyCircumcenterAngle BisectorIncenterMedianCentroidAltitudeOrthocenter

Thoerems (and how to apply them)

Midsegment Thm.Converse of the Midsegment Thm.⊥ Bisector Thm.Converse of ⊥ Bisector ThmConcurrency of ⊥ Bisector of ∆ Thm∠ Bisector ThmConverse ∠ Bisector ThmConcurrency of ∠ Bisector of ∆ ThmConcurrency of Medians of ∆ ThmConcurrency of Altitudes of ∆ ThmTriangle Inequality Theorem

Use WXY, where R, S, and T are midpoints of the sides....

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