Transcript Glencoe Geometry

Five-Minute Check (over Chapter 4)
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
Over Chapter 4
Classify the triangle.
A. scalene
B. isosceles
A. A
B. B
C. C
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B
A
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C
C. equilateral
Over Chapter 4
Find x if mA = 10x + 15, mB = 8x – 18, and
mC = 12x + 3.
A. 3.75
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B
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A
D. 16.5
A
B
C
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D
D
C. 12
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B.
C.
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D.
C
B. 6
Over Chapter 4
Name the corresponding congruent sides if
ΔRST  ΔUVW.
A. R  V, S  W, T  U
B. R  W, S  U, T  V
A. A
B. B
C. C
R  U, S  W, T  V
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B
D.
A
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C
C. R  U, S  V, T  W
Over Chapter 4
Name the corresponding congruent sides if
ΔLMN  ΔOPQ.
A.
A. A
B. B
C. C
C.
,
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B
D.
A
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C
B.
Over Chapter 4
Find y if ΔDEF is an equilateral triangle and
mF = 8y + 4.
A. 22
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B
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A
D. 4.5
A
B
C
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D
D
C. 7
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B.
C.
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D.
C
B. 10.75
Over Chapter 4
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)
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B
D. (4, –3)
A
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A
B
C
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D
D
C. (–2, 11)
C
B. (4, 0)
A.
B.
C.
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D.
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.
Content Standards
G-CO.9 Prove theorems about lines and angles.
G-CO.10 Prove theorems about triangles.
G-MG.3 Apply geometric concepts in modeling situations.
Mathematical Practices
2 Reason abstractly and quantitatively.
6 Attend to precision.
• perpendicular bisector
• concurrent lines
• point of concurrency
• circumcenter
• incenter
Use the Perpendicular Bisector Theorems
A. Find the measure of BC.
BC = AC
Perpendicular Bisector Theorem
BC = 8.5
Substitution
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 = RQ
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
Perpendicular Bisector Theorem
A. Find the measure of NO.
A. 4.6
B. 9.2
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B
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A
B
C
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D
D
D. 36.8
C
C. 18.4
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B.
C.
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D.
B. Find the measure of TU.
A. 2
B. 4
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B
A
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A
B
C
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D
D
D. 16
C
C. 8
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B.
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D.
C. Find the measure of EH.
A. 8
B. 12
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B
A
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A
B
C
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D
D
D. 20
C
C. 16
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C.
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D.
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.
BILLIARDS A 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.
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B
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A
B. Yes, circumcenter of an acute
triangle is found in the interior of
the triangle.
A. A
B. B
Use the Angle Bisector Theorems
A. Find DB.
DB = DC
Angle Bisector Theorem
DB = 5
Substitution
Answer: DB = 5
Use the Angle Bisector Theorems
B. Find WYZ.
Use the Angle Bisector Theorems
WYZ  XYZ
Definition of angle bisector
mWYZ = mXYZ
Definition of congruent angles
mWYZ = 28
Substitution
Answer: mWYZ = 28
Use the Angle Bisector Theorems
C. Find QS.
QS = SR
4x – 1 = 3x + 2
x–1 =2
x =3
Angle Bisector Theorem
Substitution
Subtract 3x from each side.
Add 1 to each side.
Answer: So, QS = 4(3) – 1 or 11.
A. Find the measure of SR.
A. 22
B. 5.5
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B
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A
B
C
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D
D
D. 2.25
C
C. 11
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B.
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B. Find the measure of HFI.
A. 28
B. 30
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B
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A
B
C
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D
D
D. 30
C
C. 15
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B.
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C. Find the measure of UV.
A. 7
B. 14
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B
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B
C
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D
D
D. 25
C
C. 19
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B.
C.
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D.
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.
Use the Incenter Theorem
Since length cannot be negative, use only the positive
square root, 6.
Answer: SU = 6
Use the Incenter Theorem
B. Find SPU if S is the incenter
of ΔMNP.
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
UPR + RMT + TNU = 180
UPR + 62 + 56 = 180
UPR + 118 = 180
UPR = 62
Triangle Angle Sum
Theorem
Substitution
Simplify.
Subtract 118 from each
side.
Since SP bisects UPR, 2mSPU = UPR. This means
1 UPR.
that mSPU = __
2
1 (62) or 31
Answer: mSPU = __
2
A. Find the measure of GF if D is the
incenter of ΔACF.
A. 12
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B
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A
D. 65
A
B
C
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D
D
C. 8
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B.
C.
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D.
C
B. 144
B. Find the measure of BCD if D
is the incenter of ΔACF.
A. 58°
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B
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A
D. 26°
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B
C
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D
D
C. 52°
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B.
C.
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D.
C
B. 116°