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
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 TheoremLesson Menu
Classify the triangle.
C. equilateral5-Minute Check 1
Find x if mA = 10x + 15, mB = 8x – 18, andmC = 12x + 3.
D. 16.55-Minute Check 2
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 V5-Minute Check 3
Name the corresponding congruent sides if ΔLMN ΔOPQ.5-Minute Check 4
Find y if ΔDEF is an equilateral triangle and mF = 8y + 4.
D. 4.55-Minute Check 5
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
A. Find the measure of BC.
BC = AC Perpendicular Bisector Theorem
BC = 8.5 Substitution
Answer: 8.5Example 1
B. Find the measure of XY.
Answer: 6Example 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: 7Example 1
A. Find the measure of NO.
D. 36.8Example 1
B. Find the measure of TU.
D. 16Example 1
C. Find the measure of EH.
D. 20Example 1
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.
Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.Example 2
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
A. Find DB.
DB = DC Angle Bisector Theorem
DB = 5 Substitution
Answer:DB = 5Example 3
B. FindWYZ.Example 3
WYZ XYZ Definition of angle bisector
mWYZ = mXYZ Definition of congruent angles
mWYZ = 28 Substitution
Answer:mWYZ = 28Example 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. Find the measure of SR.
D. 2.25Example 3
B. Find the measure of HFI.
D. 30Example 3
C. Find the measure of UV.
D. 25Example 3
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 = 6Example 4
Since 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 4
Since 1–4)SP bisects UPR, 2mSPU = UPR. This means that mSPU = UPR.
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 4
A. Find the measure of GF if D is the incenter of ΔACF.
D. 65Example 4
B. Find the measure of BCD if D is the incenter of ΔACF.
D. 26°Example 4