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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

Lesson MenuB

C

D

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

A. 3.75

B. 6

C. 12

D. 16.5

5-Minute Check 2B

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 3B

C

D

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

A. 22

B. 10.75

C. 7

D. 4.5

5-Minute Check 5B

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 6MA.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.

NGSSSYou 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.

Concept 1–4)

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 1Use 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 1Concept 1–4)

Concept 1–4)

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 2Use 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 2A 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 2Concept 1–4)

Use the Angle Bisector Theorems 1–4)

A. Find DB.

DB = DC Angle Bisector Theorem

DB = 5 Substitution

Answer:DB = 5

Example 3Use the Angle Bisector Theorems 1–4)

WYZ XYZ Definition of angle bisector

mWYZ = mXYZ Definition of congruent angles

mWYZ = 28 Substitution

Answer:mWYZ = 28

Example 3Use 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 3Concept 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 4Since length cannot be negative, use only the positive square root, 6.

Answer:SU = 6

Example 4Since 1–4)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.

Example 4Since 1–4)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 = 180 Triangle Angle Sum Theorem

UPR + 62 + 56 = 180 Substitution

UPR + 118 = 180 Simplify.

UPR = 62 Subtract 118 from each side.

Example 4A 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 4A 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 4End of the Lesson 1–4)

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