Geometry. 6.1 Perpendicular and Angle Bisectors.
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1 Geometry 6.1 Perpendicular and Angle Bisectors
2 6.1 Essential Question What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle? January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 2
3 The perpendicular bisector of a segment can be a segment. R A B RS is a bisector of AB. S January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 3
4 The perpendicular bisector of a segment can be a line. R A B RS is a bisector of AB. S January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 4
5 The perpendicular bisector of a segment can be a ray. A R B S RS is a bisector of AB. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 5
6 The perpendicular bisector of a segment can be a plane. K A B K is a bisector of AB. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 6
7 Equidistant Points A point is equidistant from two points if its distance to each point is the same. A R B R is equidistant from A and B. S is also equidistant from A and B. S January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 7
8 Theorem 6.1 (Perpendicular Bisector Theorem) AR BR If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A R S B AS BS January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 8
9 Theorem 6.2 (Converse of Perpendicular Bisector Thm.) If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. A R B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 9
10 Example 1 Find RB. 14 R 14? A B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 10
11 Example 2 Find AD. From the figure, BD is the perpendicular bisector of AC. AD = CD Perpendicular Bisector Theorem 5x = 3x + 14 Substitute. x = 7 Solve for x. So, AD = 5x = 5(7) AD = 35 January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 11
12 Your Turn 1 Find WX when WZ = 20.5, WY =14.8, and YZ = January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 12
13 Your Turn 2 ZX is the perpendicular bisector of WY, WZ = 4n 13, and YZ = n Find YZ. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 13
14 Distance from a point to a line. Defined as the length of the perpendicular segment between the point and the line. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 14
15 Distance from a point to a line. Defined as the length of the perpendicular segment between the point and the line. This is the distance from the point to the line. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 15
16 Equidistant from two lines. Point J is equidistant from lines m and n. m J n January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 16
17 Theorem 6.3 (Angle Bisector Theorem) If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. A D C B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 17
18 Theorem 6.4 (Converse of Angle Bisector Theorem) If a point is equidistant from the two sides of the angle, then it is on the bisector of an angle. A D C B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 18
19 Example 3 If AD = 15, then DC =. 15 A D C B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 19
20 Example 4 Is D on the bisector of the angle?yes 4 D 4 January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 20
21 Example 5 Is E on the bisector of the angle?no! The length of the segment from E to each is not the length of a perpendicular segment. E 5 5 January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 21
22 Example 6 Find RS. PS = RS Angle Bisector Theorem 5x = 6x 5 Substitute. 5 = x Solve for z. So, RS = 6x 5 = 6(5) 5 RS = 25 January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 22
23 Your Turn 3 Find m ABC when AD = 3.2, CD = 3.2, and m DBC = 39. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 23
24 Your Turn 4 BD bisects ABC, AD = 3z + 7, and CD = 2z Find CD. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 24
25 Summary AR BR A point on the perpendicular bisector of a segment is R equidistant from the endpoints of the segment. A B S AS BS January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 25
26 Summary A point on the bisector of an angle is equidistant from the sides of the angle. A D C B January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 26
27 Summary The distance from a point to a line is the measure of the perpendicular segment. January 5, 2017 Geometry 5.1 Perpendiculars and Bisectors 27
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