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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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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

B

C

Classify the triangle.

A. scalene

B. isosceles

C. equilateral

5-Minute Check 1

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

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

B

C

A.

B.

C.

D.

,

Name the corresponding congruent sides if ΔLMN ΔOPQ.

5-Minute Check 4

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

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

• Identify and use perpendicular bisectors in triangles.

• Identify and use angle bisectors in triangles.

Then/Now

• concurrent lines

• point of concurrency

• circumcenter

• incenter

Vocabulary

Concept 1–4)

A. Find the measure of BC.

BC = AC Perpendicular Bisector Theorem

BC = 8.5 Substitution

Answer: 8.5

Example 1

B. Find the measure of XY.

Answer: 6

Example 1

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

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

A 1–4)

B

C

D

B. Find the measure of TU.

A. 2

B. 4

C. 8

D. 16

Example 1

A 1–4)

B

C

D

C. Find the measure of EH.

A. 8

B. 12

C. 16

D. 20

Example 1

Concept 1–4)

Concept 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

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 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

Concept 1–4)

A. Find DB.

DB = DC Angle Bisector Theorem

DB = 5 Substitution

Answer:DB = 5

Example 3

B. FindWYZ.

Example 3

WYZ  XYZ Definition of angle bisector

mWYZ = mXYZ Definition of congruent angles

mWYZ = 28 Substitution

Answer:mWYZ = 28

Example 3

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

A 1–4)

B

C

D

A. Find the measure of SR.

A. 22

B. 5.5

C. 11

D. 2.25

Example 3

A 1–4)

B

C

D

B. Find the measure of HFI.

A. 28

B. 30

C. 15

D. 30

Example 3

A 1–4)

B

C

D

C. Find the measure of UV.

A. 7

B. 14

C. 19

D. 25

Example 3

Concept 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

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

Answer:SU = 6

Example 4

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

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

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

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