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Lesson Menu Five-Minute Check (over Chapter 4) CCSS 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 measure angle A = 10x + 15, measure angle B = 8x – 18, and measure angle C = 12x + 3.

Over Chapter 4 5-Minute Check 4 Name the corresponding congruent sides if ΔLMN is congruent to 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 measure of Angle F = 8y + 4.

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

Vocabulary perpendicular bisector a segment which bisects a line segment into two equal parts at 90° concurrent lines point of concurrency circumcenter incenter

Vocabulary perpendicular bisector concurrent lines three or more lines in a plane or higher- dimensional space that intersect at a single point. point of concurrency circumcenter incenter

Vocabulary perpendicular bisector concurrent lines point of concurrency the point where three or more lines intersect circumcenter incenter

Vocabulary perpendicular bisector concurrent lines point of concurrency circumcenter the point where the perpendicular bisectors of the sides intersect. incenter

Vocabulary perpendicular bisector concurrent lines point of concurrency Circumcenter incenter a triangle’s center

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 measure angle WYZ.

Example 3 Use the Angle Bisector Theorems Answer: m WYZ = 28 WYZ ≅ XYWDefinition of angle bisector m WYZ= m XYWDefinition of congruence 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 ST 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: ST = 6 Since length cannot be negative, use only the positive square root, 6. Since ST = SU, ST = 6. 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.

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Over Lesson 5–1 5-Minute Check 1 A.–5 B.0.5 C.5 D.10 In the figure, A is the circumcenter of ΔLMN. Find y if LO = 8y + 9 and ON = 12y – 11.

Over Lesson 5–1 5-Minute Check 2 A.13 B.11 C.7 D.–13 In the figure, A is the circumcenter of ΔLMN. Find x if m APM = 7x + 13.

Over Lesson 5–1 5-Minute Check 3 A.–12.5 B.2.5 C D.12.5 In the figure, A is the circumcenter of ΔLMN. Find r if AN = 4r – 8 and AM = 3(2r – 11).

Over Lesson 5–1 5-Minute Check 4 In the figure, point D is the incenter of ΔABC. What segment is congruent to DG? ___ A.DE B.DA C.DC D.DB ___

Over Lesson 5–1 5-Minute Check 5 A. GCD B. DCG C. DFB D. ADE In the figure, point D is the incenter of ΔABC. What angle is congruent to DCF?

Over Lesson 5–1 5-Minute Check 6 A.It is equidistant from the sides of the triangle. B.It can be located outside of the triangle. C.It is the point where the perpendicular bisectors intersect. D.It is the center of the circumscribed circle. Which of the following statements about the circumcenter of a triangle is false?

Then/Now You identified and used perpendicular and angle bisectors in triangles. Identify and use medians in triangles. Identify and use altitudes in triangles.

Vocabulary median centroid altitude orthocenter

Vocabulary Median a line segment joining a vertex to the midpoint of the opposing side centroid altitude orthocenter

Vocabulary median Centroid the point where the three medians of the triangle meet. altitude orthocenter

Vocabulary median centroid Altitude a line segment through a vertex and perpendicular to an opposite line segment orthocenter

Vocabulary median centroid altitude Orthocenter Point where three altitudes intersect in a single point. (The orthocenter lies inside the triangle if and only if the triangle is acute)

Concept

Example 1 Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify.

Example 1 Use the Centroid Theorem Answer: YP = 8; PV = 4 YP + PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each side.

Example 1 A.LR = 15; RO = 15 B.LR = 20; RO = 10 C.LR = 17; RO = 13 D.LR = 18; RO = 12 In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

Example 2 Use the Centroid Theorem In ΔABC, CG = 4. Find GE.

Example 2 Use the Centroid Theorem Centroid Theorem CG = 4 6 = CE

Example 2 Use the Centroid Theorem Answer: GE = 2 Segment AdditionCG + GE = CE Substitution4 + GE = 6 Subtract 4 from each side.GE = 2

Example 2 A.4 B.6 C.16 D.8 In ΔJLN, JP = 16. Find PM.

Example 3 Find the Centroid on a Coordinate Plane SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance? UnderstandYou need to find the centroid of the triangle. This is the point at which the triangle will balance.

Example 3 Find the Centroid on a Coordinate Plane SolveGraph the triangle and label the vertices A, B, and C. PlanGraph and label the triangle with vertices at (1, 4), (3, 0), and (3, 8). Use the Midpoint Theorem to find the midpoint of one of the sides of the triangle. The centroid is two-thirds the distance from the opposite vertex to that midpoint.

Example 3 Find the Centroid on a Coordinate Plane Graph point D. Find the midpoint D of BC.

Example 3 Find the Centroid on a Coordinate Plane Notice that is a horizontal line. The distance from D(3, 4) to A(1, 4) is 3 – 1 or 2 units.

The centroid P is the distance. So, the centroid is (2) or units to the right of A. The coordinates are. Example 3 Find the Centroid on a Coordinate Plane P

Example 3 Find the Centroid on a Coordinate Plane Answer: The artist should place the pole at the point CheckCheck the distance of the centroid from point D(3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid is. __

Example 3 BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at (–3, 2), (–1, –2), and (–1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance? A. B. C. (–1, 2)D. (0, 4)

Concept

Example 4 Find the Orthocenter on a Coordinate Plane COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ.

Example 4 Find the Orthocenter on a Coordinate Plane Find an equation of the altitude from The slope of so the slope of an altitude is Point-slope form Distributive Property Add 1 to each side.

Example 4 Find the Orthocenter on a Coordinate Plane Point-slope form Distributive Property Subtract 3 from each side. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6.

Example 4 Find the Orthocenter on a Coordinate Plane Equation of altitude from J Multiply each side by 5. Add 105 to each side. Add 4x to each side. Divide each side by –26. Substitution, Then, solve a system of equations to find the point of intersection of the altitudes.

Example 4 Find the Orthocenter on a Coordinate Plane Replace x with in one of the equations to find the y-coordinate. Multiply and simplify. Rename as improper fractions.

Example 4 Find the Orthocenter on a Coordinate Plane Answer: The coordinates of the orthocenter of ΔHIJ are

Example 4 A.(1, 0) B.(0, 1) C.(–1, 1) D.(0, 0) COORDINATE GEOMETRY The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC.

Concept

End of the Lesson