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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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### 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

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

A

B

C

Classify the triangle.

A.scalene

B.isosceles

C.equilateral

### 5-Minute Check 1

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

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

A

B

C

A.

B.

C.

D.

,

Name the corresponding congruent sides if ΔLMN ΔOPQ.

### 5-Minute Check 4

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

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

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

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

• perpendicular bisector

• concurrent lines

• point of concurrency

• circumcenter

• incenter

### Concept

Use the Perpendicular Bisector Theorems

A. Find the measure of BC.

BC= ACPerpendicular Bisector Theorem

BC= 8.5Substitution

### Example 1

Use the Perpendicular Bisector Theorems

B. Find the measure of XY.

### Example 1

Use the Perpendicular Bisector Theorems

C. Find the measure of PQ.

PQ= RQPerpendicular Bisector Theorem

3x + 1= 5x – 3Substitution

1= 2x – 3Subtract 3x from each side.

4= 2xAdd 3 to each side.

2= xDivide each side by 2.

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

### Example 1

A

B

C

D

A. Find the measure of NO.

A.4.6

B.9.2

C.18.4

D.36.8

### Example 1

A

B

C

D

B. Find the measure of TU.

A.2

B.4

C.8

D.16

### Example 1

A

B

C

D

C. Find the measure of EH.

A.8

B.12

C.16

D.20

### Concept

Use the Circumcenter Theorem

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

Use the Circumcenter Theorem

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

A

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.

### Concept

Use the Angle Bisector Theorems

A. Find DB.

DB= DCAngle Bisector Theorem

DB= 5Substitution

### Example 3

Use the Angle Bisector Theorems

B. FindWYZ.

### Example 3

Use the Angle Bisector Theorems

WYZ XYZDefinition of angle bisector

mWYZ= mXYZDefinition of congruent angles

mWYZ= 28Substitution

### Example 3

Use the Angle Bisector Theorems

C. Find QS.

QS= SRAngle Bisector Theorem

4x – 1= 3x + 2Substitution

x – 1= 2Subtract 3x from each side.

x= 3Add 1 to each side.

Answer: So, QS = 4(3) – 1 or 11.

### Example 3

A

B

C

D

A. Find the measure of SR.

A.22

B.5.5

C.11

D.2.25

### Example 3

A

B

C

D

B. Find the measure of HFI.

A.28

B.30

C.15

D.30

### Example 3

A

B

C

D

C. Find the measure of UV.

A.7

B.14

C.19

D.25

### Concept

Use the Incenter Theorem

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

Find SU by using the Pythagorean Theorem.

a2 + b2= c2Pythagorean Theorem

82 + SU2= 102Substitution

64 + SU2= 10082 = 64, 102 = 100

SU2= 36Subtract 64 from each side.

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

### Example 4

Use the Incenter Theorem

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

### Example 4

Since 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

Since SP bisects UPR, 2mSPU = UPR. This means that mSPU = UPR.

1

1

__

__

2

2

Use the Incenter Theorem

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

UPR + 62 + 56= 180Substitution

UPR + 118= 180Simplify.

UPR = 62Subtract 118 from each side.

### Example 4

A

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

A

B

C

D

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

A.58°

B.116°

C.52°

D.26°