Bisectors of Triangles LESSON 5–1

Lesson Menu Five-Minute Check (over Chapter 4) TEKS 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 5-Minute Check 1 A.scalene B.isosceles C.equilateral Classify the triangle.

Over Chapter 4 5-Minute Check 2 A.3.75 B.6 C.12 D.16.5 Find x if m  A = 10x + 15, m  B = 8x – 18, and m  C = 12x + 3.

Over Chapter 4 5-Minute Check 3 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.

Over Chapter 4 5-Minute Check 4 Name the corresponding congruent sides if ΔLMN  ΔOPQ. A. B. C. D.,

Over Chapter 4 5-Minute Check 5 A.22 B C.7 D.4.5 Find y if ΔDEF is an equilateral triangle and m  F = 8y + 4.

Over Chapter 4 5-Minute Check 6 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?

TEKS Targeted TEKS G.6(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems. G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(G)

Then/Now You used segment and angle bisectors. Identify and use perpendicular bisectors in triangles. Identify and use angle bisectors in triangles.

Vocabulary perpendicular bisector concurrent lines point of concurrency circumcenter incenter

Concept

Example 1 Use the Perpendicular Bisector Theorems A. Find BC. Answer: 8.5 BC= ACPerpendicular Bisector Theorem BC= 8.5Substitution

Example 1 Use the Perpendicular Bisector Theorems B. Find XY. Answer: 6

Example 1 Use the Perpendicular Bisector Theorems C. Find 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

Example 1 A.4.6 B.9.2 C.18.4 D.36.8 A. Find NO.

Example 1 A.2 B.4 C.8 D.16 B. Find TU.

Example 1 A.8 B.12 C.16 D.20 C. Find EH.

Concept

Example 2 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.

Example 2 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

Example 2 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?

Concept

Example 3 Use the Angle Bisector Theorems A. Find DB. Answer: DB = 5 DB= DCAngle Bisector Theorem DB= 5Substitution

Example 3 Use the Angle Bisector Theorems B. Find m  WYZ.

Example 3 Use the Angle Bisector Theorems Answer: m  WYZ = 28  WYZ   XYWDefinition of angle bisector m  WYZ= m  XYWDefinition of congruent angles m  WYZ= 28Substitution

Example 3 Use the Angle Bisector Theorems C. Find QS. Answer: So, QS = 4(3) – 1 or 11. QS= SRAngle Bisector Theorem 4x – 1= 3x + 2Substitution x – 1= 2Subtract 3x from each side. x= 3Add 1 to each side.

Example 3 A.22 B.5.5 C.11 D.2.25 A. Find the measure of SR.

Example 3 A.28 B.30 C.15 D.30 B. Find the measure of  HFI.

Example 3 A.7 B.14 C.19 D.25 C. Find the measure of UV.

Concept

Example 4 Use the Incenter Theorem A. Find ST if S is the incenter of ΔMNP. By the Incenter Theorem, since S is equidistant from the sides of ΔMNP, ST = SU. Find SU by using the Pythagorean Theorem. a 2 + b 2 = c 2 Pythagorean Theorem SU 2 = 10 2 Substitution 64 + SU 2 = = 64, 10 2 = 100

Example 4 Use the Incenter Theorem Answer: SU = 6 Since length cannot be negative, use only the positive square root. SU 2 = 36Subtract 64 from each side. SU= ±6Take the square root of each side.

Example 4 Use the Incenter Theorem B. Find m  SPU if S is the incenter of ΔMNP. Since MS bisects  RMT, m  RMT = 2m  RMS. So m  RMT = 2(31) or 62. Likewise, m  TNU = 2m  SNU, so m  TNU = 2(28) or 56.

Example 4 Use the Incenter Theorem m  UPR + m  RMT + m  TNU =180Triangle Angle Sum Theorem m  UPR =180Substitution m  UPR =180Simplify. m  UPR =62Subtract 118 from each side. Since PS bisects  UPR, 2m  SPU = m  UPR. This means that m  SPU = m  UPR. __ 1 2 Answer: m  SPU = (62) or 31 __ 1 2

Example 4 A.12 B.144 C.8 D.65 A. Find the measure of GF if D is the incenter of ΔACF.

Example 4 A.58° B.116° C.52° D.26° B. Find the measure of  BCD if D is the incenter of ΔACF.

Bisectors of Triangles LESSON 5–1