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Equidistant. A point is equidistant from two figures if the point is the same distance from each figure. Examples: midpoints and parallel lines. 5.2-5.4: Special Segments. Objectives: To use and define perpendicular bisectors, angle bisectors, medians, and altitudes - PowerPoint PPT Presentation
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Equidistant
A point is equidistant from two figures if the point is the same distance from each figure.
Examples: midpoints and parallel lines
5.2-5.4: Special Segments
Objectives:
1. To use and define perpendicular bisectors, angle bisectors, medians, and altitudes
2. To discover, use, and prove various theorems about perpendicular bisectors and angle bisectors
Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse of Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Example 1
Plan a proof for the Perpendicular Bisector Theorem.
1. AB CP Given2. AP = PB Def. of Bisector 3. <CPB & Def. of Perp. <CPA R rt. <s 4. <CPB=<CPA All rt. <s R =5. CP = CP Reflexive6. CAP = CBP SAS7. CA = CB CPCTC
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Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Example 4
A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goal post or the left one? Answer in your notebook
Example 6
Find the measure of <GFJ.
It’s not the Angle Bisector Theorem that could help us answer this question. It’s the converse. If it’s true.
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
Special Triangle Segments
Both perpendicular bisectors and angle bisectors are often associated with triangles, as shown below. Triangles have two other special segments.
A
B
C
B
A C
Angle Bisector
Pe
rpe
nd
icu
lar
Bis
ec
tor
Median
A median of a triangle is a segment from a vertex to the midpoint of the opposite side of the triangle.
Altitude
The length of the altitude is the height of the triangle.
An altitude of a triangle is a perpendicular segment from a vertex to the opposite side or to the line that contains that side.
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