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CHAPTER 5.1 Check.4.14 Identify and use medians, mid-segments, altitudes, angle bisectors, and perpendicular bisectors of triangles to solve problems (e.g., find segment lengths, angle measures, points of concurrency). POINTS OF CONCURRENCY The point where the Perpendicular bisectors of a triangle meet is called the _______________. The point where the Angle Bisectors of a triangle meet is called the _______________. The ___________of a triangle is equidistant from the __________of the triangle. The point where the Medians of a triangle meet is called the _______________. The __________ of a triangle is ___________from the of the _________triangle. The ________of a triangle is located _________of the distance from a _________ to the _________ of the side opposite the vertex. The point where the Altitudes of a triangle meet is called the _______________. PERPENDICULAR BISECTORS - THEOREMS Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. ABCD and AB bisects CD then AC = AD and BC = BD A D C B PERPENDICULAR BISECTORS - THEOREMS Converse Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment If AC = AD, then A lies on the perpendicular bisector of CD. If BC = BD then B lies on the perpendicular bisector of CD. A D C B USE THE PERPENDICULAR BISECTOR THEOREMS A. Find BC. BC = AC Perpendicular Bisector Theorem BC = 8.5 Substitution Answer: 8.5 USE THE PERPENDICULAR BISECTOR THEOREMS B. Find XY. Answer: 6 CIRCUMCENTER THEOREM The point where the Perpendicular bisectors of a triangle meet. B AJ = CJ AJ = BJ CJ = BJ Transitive property J A Circumcenter of a triangle is equidistant from the vertices of the triangle. C POINTS OF CONCURRENCY The point where the Perpendicular bisectors of a Circumcenter triangle meet is called the _______________. The point where the Angle Bisectors of a triangle Incenter meet is called the _______________. The ___________of a triangle is equidistant from the __________of the triangle. The point where the Medians of a triangle meet is Centroid called the _______________. Circumcenter equidistant The __________ of a triangle is ___________from the of the _________triangle. verticies The ________of a triangle is located _________of the distance from a _________ to the _________ of the side opposite the vertex. The point where the Altitudes of a triangle meet is called the _______________. Orthocenter ANGLE BISECTORS Any point on the angle bisector is equidistant from the sides of the angle. B J Converse A Any point equidistant from the sides of an angle lies on the angle bisector C 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 mWYZ. USE THE ANGLE BISECTOR THEOREMS WYZ XYW Definition of angle bisector mWYZ = mXYW Definition of congruent angles mWYZ = 28 Substitution Answer: mWYZ = 28 INCENTER THEOREM The point where the Angle Bisectors of a triangle meet. B J A The incenter of a triangle is equidistant from the sides of the triangle. C POINTS OF CONCURRENCY The point where the Perpendicular bisectors of a triangle meet is called the _______________. Circumcenter The point where the Angle Bisectors of a triangle Incenter meet is called the _______________. The ___________of incenter a triangle is equidistant from the sides __________of the triangle. The point where the Medians of a triangle meet is Centroid called the _______________. TheCircumcenter __________ of a triangle is equidistant ___________from the of the _________triangle. verticies The ________of a triangle is located _________of the distance from a _________ to the _________ of the side opposite the vertex. The point where the Altitudes of a triangle meet is called the _______________. Orthocenter 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. 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. Since length cannot be negative, use only the positive square root, 6. Since ST = SU, ST = 6. Answer: ST = 6 The Centroid is the point of balance for any triangle. CENTROID THEOREM The point where the Medians of a triangle meet. B Median connects the Midpoint of the side with the vertex of the opposite angle D A E L F The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex. AL= 2/3 AE, BL=2/3BF and DL=2/3DC C POINTS OF CONCURRENCY The point where the Perpendicular bisectors of a triangle meet is called the Circumcenter _______________. The point where the Angle Bisectors of a triangle Incenter meet is called the _______________. The ___________of incenter a triangle is equidistant from the sides __________of the triangle. The point where the Medians of a triangle meet is Centroid called the _______________. equidistant TheCircumcenter __________ of a triangle is ___________from the of the _________triangle. verticies 2/3 Centroid The ________of a triangle is located ____of the distance vertex midpoint from a __________ to the ___________ of the side opposite the vertex. The point where the Altitudes of a triangle meet is called the _______________. Orthocenter USE THE CENTROID THEOREM In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify. YP + PV = YV 8 + PV = 12 PV =4 Segment Addition Subtract 8 from each side Answer: YP = 8; PV = 4 Points S, T, and U are the midpoint of DE, EF and DF respectively. Find x, y, and z Centroid Theorem CALCULATIONS E S 6 D y T 4z A 2x-5 2.9 U 4.6 F Centroid is located at 2/3 the length of the median EA=2/3EU DT=DA+AT Y=2/3(y+2.9) DT=2x-5+6=2x+1 3Y = 2y + 5.8 1y=5.8 DA=2/3DT 6=2/3(2x+1) FA=2/3FS 3/2(6)=2x+1 4.6=2/3(4z+4.6) 9=2x+1 13.8=8z + 9.2 X=4 4.6 = 8z Z=0.575 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) FIND THE ORTHOCENTER OF JKL Find Altitude from J to KL LK Slope LK is -1/3, m= 3 Point slope (1,3) m= 3 y-3=3(x-1) y-3=3x-3 y = 3x Find Altitude from K to JL Slope of JL is 3/2, m=-2/3 y+1=-2/3(x-2) y+1=-2/3x+4/3 y = -2/3x+1/3 J(1,3) L(-1,0) K(2,-1) y = 3x y = -2/3x+1/3 find intersection 3x= -2/3x+1/3 multiply by 3 9x =-2x + 1 11x = 1 x = 1/11 y=3(1/11) y=3/11 PRACTICE ASSIGNMENT Page 327, 10 – 30 4th, Page 338, 6 – 30 4th