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Splash Screen. Five-Minute Check (over Chapter 4) NGSSS Then/Now New Vocabulary Theorems: Perpendicular Bisectors Example 1: Use the Perpendicular Bisector Theorems Theorem 5.3: Circumcenter Theorem Proof: Circumcenter Theorem

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Lesson menu

Five-Minute Check (over Chapter 4)

NGSSS

Then/Now

New Vocabulary

Theorems: Perpendicular Bisectors

Example 1: Use the Perpendicular Bisector Theorems

Theorem 5.3: Circumcenter Theorem

Proof: Circumcenter Theorem

Example 2: Real-World Example: Use the Circumcenter Theorem

Theorems: Angle Bisectors

Example 3: Use the Angle Bisector Theorems

Theorem 5.6: Incenter Theorem

Example 4: Use the Incenter Theorem

Lesson Menu


5 minute check 1

A

B

C

Classify the triangle.

A. scalene

B. isosceles

C. equilateral

5-Minute Check 1


5 minute check 2

A

B

C

D

Find x if mA = 10x + 15, mB = 8x – 18, andmC = 12x + 3.

A. 3.75

B. 6

C. 12

D. 16.5

5-Minute Check 2


5 minute check 3

A

B

C

Name the corresponding congruent sides if ΔRST ΔUVW.

A. R  V,S  W,T  U

B. R  W,S  U,T  V

C. R  U,S  V,T  W

D. R  U,S  W,T  V

5-Minute Check 3


5 minute check 4

A

B

C

A.

B.

C.

D.

,

Name the corresponding congruent sides if ΔLMN ΔOPQ.

5-Minute Check 4


5 minute check 5

A

B

C

D

Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.

A. 22

B. 10.75

C. 7

D. 4.5

5-Minute Check 5


5 minute check 6

A

B

C

D

ΔABC has vertices A(–5, 3) and B(4, 6). What are the coordinates for point C if ΔABC is an isosceles triangle with vertex angle A?

A. (–3, –6)

B. (4, 0)

C. (–2, 11)

D. (4, –3)

5-Minute Check 6


Ngsss

MA.912.G.4.1Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.

MA.912.G.4.2Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter.

NGSSS


Then now

You used segment and angle bisectors. (Lesson 1–3 and 1–4)

  • Identify and use perpendicular bisectors in triangles.

  • Identify and use angle bisectors in triangles.

Then/Now


Vocabulary

  • concurrent lines

  • point of concurrency

  • circumcenter

  • incenter

Vocabulary


Concept
Concept 1–4)


Example 1

Use the Perpendicular Bisector Theorems 1–4)

A. Find the measure of BC.

BC = AC Perpendicular Bisector Theorem

BC = 8.5 Substitution

Answer: 8.5

Example 1


Example 11

Use the Perpendicular Bisector Theorems 1–4)

B. Find the measure of XY.

Answer: 6

Example 1


Example 12

Use the Perpendicular Bisector Theorems 1–4)

C. Find the measure of PQ.

PQ = RQ Perpendicular Bisector Theorem

3x + 1 = 5x – 3 Substitution

1 = 2x – 3 Subtract 3x from each side.

4 = 2x Add 3 to each side.

2 = x Divide each side by 2.

So, PQ = 3(2) + 1 = 7.

Answer: 7

Example 1


Example 13

A 1–4)

B

C

D

A. Find the measure of NO.

A. 4.6

B. 9.2

C. 18.4

D. 36.8

Example 1


Example 14

A 1–4)

B

C

D

B. Find the measure of TU.

A. 2

B. 4

C. 8

D. 16

Example 1


Example 15

A 1–4)

B

C

D

C. Find the measure of EH.

A. 8

B. 12

C. 16

D. 20

Example 1


Concept1
Concept 1–4)


Concept2
Concept 1–4)


Example 2

Use the Circumcenter Theorem 1–4)

GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?

By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.

Example 2


Example 21

Use the Circumcenter Theorem 1–4)

Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle.

C

Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.

Example 2


Example 22

A 1–4)

B

BILLIARDSA triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle?

A. No, the circumcenter of an acute triangle is found in the exterior of the triangle.

B. Yes, circumcenter of an acute triangle is found in the interior of the triangle.

Example 2


Concept3
Concept 1–4)


Example 3

Use the Angle Bisector Theorems 1–4)

A. Find DB.

DB = DC Angle Bisector Theorem

DB = 5 Substitution

Answer:DB = 5

Example 3


Example 31

Use the Angle Bisector Theorems 1–4)

B. FindWYZ.

Example 3


Example 32

Use the Angle Bisector Theorems 1–4)

WYZ  XYZ Definition of angle bisector

mWYZ = mXYZ Definition of congruent angles

mWYZ = 28 Substitution

Answer:mWYZ = 28

Example 3


Example 33

Use the Angle Bisector Theorems 1–4)

C. Find QS.

QS = SR Angle Bisector Theorem

4x – 1 = 3x + 2 Substitution

x – 1 = 2 Subtract 3x from each side.

x = 3 Add 1 to each side.

Answer: So, QS = 4(3) – 1 or 11.

Example 3


Example 34

A 1–4)

B

C

D

A. Find the measure of SR.

A. 22

B. 5.5

C. 11

D. 2.25

Example 3


Example 35

A 1–4)

B

C

D

B. Find the measure of HFI.

A. 28

B. 30

C. 15

D. 30

Example 3


Example 36

A 1–4)

B

C

D

C. Find the measure of UV.

A. 7

B. 14

C. 19

D. 25

Example 3


Concept4
Concept 1–4)


Example 4

Use the Incenter Theorem 1–4)

A. Find SU if S is the incenter of ΔMNP.

Find SU by using the Pythagorean Theorem.

a2 + b2 = c2 Pythagorean Theorem

82 + SU2 = 102 Substitution

64 + SU2 = 100 82 = 64, 102 = 100

SU2 = 36 Subtract 64 from each side.

SU = ±6 Take the square root of each side.

Example 4


Example 41

Use the Incenter Theorem 1–4)

Since length cannot be negative, use only the positive square root, 6.

Answer:SU = 6

Example 4


Example 42

Since 1–4)MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, TNU = 2mSNU, so mTNU = 2(28) or 56.

Use the Incenter Theorem

B. FindSPU if S is the incenter of ΔMNP.

Example 4


Example 43

Since 1–4)SP bisects UPR, 2mSPU = UPR. This means that mSPU = UPR.

1

1

__

__

2

2

Answer:mSPU = (62) or 31

Use the Incenter Theorem

UPR + RMT + TNU = 180 Triangle Angle Sum Theorem

UPR + 62 + 56 = 180 Substitution

UPR + 118 = 180 Simplify.

UPR = 62 Subtract 118 from each side.

Example 4


Example 44

A 1–4)

B

C

D

A. Find the measure of GF if D is the incenter of ΔACF.

A. 12

B. 144

C. 8

D. 65

Example 4


Example 45

A 1–4)

B

C

D

B. Find the measure of BCD if D is the incenter of ΔACF.

A. 58°

B. 116°

C. 52°

D. 26°

Example 4