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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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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Over Chapter 4
A. scalene
B. isosceles
C. equilateral
Classify the triangle.
A. A
B. B
C. C
A B C
0% 0%0%
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Over Chapter 4
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 3.75
B. 6
C. 12
D. 16.5
Find x if mA = 10x + 15, mB = 8x – 18, and mC = 12x + 3.
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Over Chapter 4
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
Name the corresponding congruent sides if ΔRST ΔUVW.
A. A
B. B
C. C
A B C
0% 0%0%
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Over Chapter 4
Name the corresponding congruent sides if ΔLMN ΔOPQ.
A. A
B. B
C. C
A B C
0% 0%0%
A.
B.
C.
D. ,
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Over Chapter 4
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 22
B. 10.75
C. 7
D. 4.5
Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.
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Over Chapter 4
A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. (–3, –6)
B. (4, 0)
C. (–2, 11)
D. (4, –3)
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?
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MA.912.G.4.1 Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.
MA.912.G.4.2 Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter.
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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.
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• perpendicular bisector
• concurrent lines
• point of concurrency
• circumcenter
• incenter
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Use the Perpendicular Bisector Theorems
A. Find the measure of BC.
Answer: 8.5
BC = AC Perpendicular Bisector Theorem
BC = 8.5 Substitution
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Use the Perpendicular Bisector Theorems
B. Find the measure of XY.
Answer: 6
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Use the Perpendicular Bisector Theorems
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
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 4.6
B. 9.2
C. 18.4
D. 36.8
A. Find the measure of NO.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 2
B. 4
C. 8
D. 16
B. Find the measure of TU.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 8
B. 12
C. 16
D. 20
C. Find the measure of EH.
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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.
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Use the Circumcenter Theorem
Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.
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
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A. A
B. B
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.
BILLIARDS A triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle?
A B
0%0%
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Use the Angle Bisector Theorems
A. Find DB.
Answer: DB = 5
DB = DC Angle Bisector Theorem
DB = 5 Substitution
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Use the Angle Bisector Theorems
B. Find WYZ.
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Use the Angle Bisector Theorems
Answer: mWYZ = 28
WYZ XYZ Definition of angle bisector
mWYZ = mXYZ Definition of congruent angles
mWYZ = 28 Substitution
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Use the Angle Bisector Theorems
C. Find QS.
Answer: So, QS = 4(3) – 1 or 11.
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.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 22
B. 5.5
C. 11
D. 2.25
A. Find the measure of SR.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 28
B. 30
C. 15
D. 30
B. Find the measure of HFI.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 7
B. 14
C. 19
D. 25
C. Find the measure of UV.
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Use the Incenter Theorem
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.
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Use the Incenter Theorem
Answer: SU = 6
Since length cannot be negative, use only the positive square root, 6.
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Use the Incenter Theorem
B. Find SPU if S is the incenter of ΔMNP.
Since MS bisects RMT, mRMT = 2mRMS. So mRMT = 2(31) or 62. Likewise, TNU = 2mSNU, so mTNU = 2(28) or 56.
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Use the Incenter Theorem
UPR + RMT + TNU = 180 Triangle Angle Sum Theorem
UPR + 62 + 56 = 180 SubstitutionUPR + 118 = 180 Simplify.
UPR = 62 Subtract 118 from each side.
Since SP bisects UPR, 2mSPU = UPR. This means
that mSPU = UPR. __12
Answer: mSPU = (62) or 31__12
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 12
B. 144
C. 8
D. 65
A. Find the measure of GF if D is the incenter of ΔACF.
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A. A
B. B
C. C
D. D A B C D
0% 0%0%0%
A. 58°
B. 116°
C. 52°
D. 26°
B. Find the measure of BCD if D is the incenter of ΔACF.
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