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Warm-Up: EOC Prep What is RS? What is m<ABC? Bisectors in Triangles 5.2 Today’s Goals By the end of class today, YOU should be able to… 1. Define and use the properties of perpendicular bisectors and angle bisectors to solve for unknowns. 2. Locate places equidistant from two given points on a map. Review… We learned in chapter 4 that ΔCAD ≅ ΔCBD. Therefore, we can conclude that CA ≅ CB, that CA = CB, or simply that C is equidistant from points A and B. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Ex.1: Using the Perpendicular Bisector Theorem If CD is the perpendicular bisector of both XY and ST, and CY = 16. Find the length of TY. Ex.1: Solution CS = CT CY – CT = TY We know from the Perpendicular Bisector Theorem that CS is equivalent to CT Subtract to find the value of TY You Try… If CD is the perpendicular bisector of both XY and ST, and CY = 16. Find the length of CX. Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. Ex.2: Using the Angle Bisector Theorem Find the value of x, then find FD and FB: Ex.2: Solution From the diagram we see that F is on the bisector of ACE. Therefore, FB = FD. FB = FD 5x = 2x + 24 Substitute 3x = 24 Subtract 2x x=8 Divide by 3 FB = 5x = 5(8) = 40 Substitute FD = 40 Substitute You Try… Does point A lie on an angle bisector of <TXR? Explain. Angle bisector constructions Construct an angle ABC Place the compass point on the angle vertex with the compass set to any convenient width Draw an arc that falls across both legs of the angle *The compass can then be adjusted at this point if desired From where an arc crosses a leg, make an arc in the angle's interior, then without changing the compass width, repeat for the other leg Draw a straight line from B to point D, where the arcs cross Done. The line just drawn bisects the angle ABC Practice A mysterious map has come into your possession. The map shows the Sea Islands off the coast of Georgia. But that’s not all! The map also contains three clues that tell where a treasure is supposedly buried! The 1st clue: Draw a line from Baxley to Savannah. From Savannah, draw a southwesterly line that forms a 60° angle with the first line. The treasure is on an island that lies along the second line. On which islands could the treasure be buried? The 2nd clue: The treasure is on an island 22 miles from Everett. Construct a figure that contains all the points 22 miles from Everett. According to the first two clues, on which islands could the treasure be buried? Explain. The 3rd clue: The perpendicular bisector of the line segment between Baxley and Jacksonville passes through the island. The treasure is buried by the lighthouse on that island. On which island is the treasure buried? Explain. Homework Page 251 #s 8-11, 16, 18 Page 252 # 30 The assignment can also be found at: • http://www.pearsonsuccessnet.com/snpap p/iText/products/0-13-037878-X/Ch05/0502/PH_Geom_ch05-02_Ex.pdf