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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

A

B

C

D

Find x if mA = 10x + 15, mB = 8x – 18, andmC = 12x + 3.

A.3.75

B.6

C.12

D.16.5

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

A

B

C

A.

B.

C.

D.

,

Name the corresponding congruent sides if ΔLMN ΔOPQ.

A

B

C

D

Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.

A.22

B.10.75

C.7

D.4.5

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)

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.

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.

- perpendicular bisector

- concurrent lines
- point of concurrency
- circumcenter
- incenter

Use the Perpendicular Bisector Theorems

A. Find the measure of BC.

BC= ACPerpendicular Bisector Theorem

BC= 8.5Substitution

Answer: 8.5

Use the Perpendicular Bisector Theorems

B. Find the measure of XY.

Answer: 6

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.

Answer: 7

A

B

C

D

A. Find the measure of NO.

A.4.6

B.9.2

C.18.4

D.36.8

A

B

C

D

B. Find the measure of TU.

A.2

B.4

C.8

D.16

A

B

C

D

C. Find the measure of EH.

A.8

B.12

C.16

D.20

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.

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.

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.

Use the Angle Bisector Theorems

A. Find DB.

DB= DCAngle Bisector Theorem

DB= 5Substitution

Answer:DB = 5

Use the Angle Bisector Theorems

B. FindWYZ.

Use the Angle Bisector Theorems

WYZ XYZDefinition of angle bisector

mWYZ= mXYZDefinition of congruent angles

mWYZ= 28Substitution

Answer:mWYZ = 28

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.

A

B

C

D

A. Find the measure of SR.

A.22

B.5.5

C.11

D.2.25

A

B

C

D

B. Find the measure of HFI.

A.28

B.30

C.15

D.30

A

B

C

D

C. Find the measure of UV.

A.7

B.14

C.19

D.25

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.

Use the Incenter Theorem

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

Answer:SU = 6

Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, TNU = 2mSNU, so mTNU = 2(28) or 56.

Use the Incenter Theorem

B. FindSPU if S is the incenter of ΔMNP.

Since SP bisects UPR, 2mSPU = UPR. This means that mSPU = UPR.

1

1

__

__

2

2

Answer:mSPU = (62) or 31

Use the Incenter Theorem

UPR + RMT + TNU= 180Triangle Angle Sum Theorem

UPR + 62 + 56= 180Substitution

UPR + 118= 180Simplify.

UPR = 62Subtract 118 from each side.

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

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°