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Vocabulary Equidistant Locus Perpendicular Bisector Angle Bisector

When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. The angle bisector of an angle is the locus of points that are equidistant from the sides of the angle.

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6.1 6.2

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Example 1A: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure. MN

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Example 1B: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure. BC

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Example 1C: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure. TU

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Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. Describe the locus of points that are equidistant from the sides of an angle.

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6.3 6.4

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Example 2A: Applying the Angle Bisector Theorem

Find the measure. BC Given that YW bisects XYZ and WZ = 3.05, find WX.

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Example 4: Writing Equations of Bisectors in the Coordinate Plane

Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1). Step 1 Graph .

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Check It Out! Example 4 Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4). Step 1 Graph PQ. The perpendicular bisector of is perpendicular to at its midpoint.

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Lesson Quiz: Part I Use the diagram for Items 1–2. 1. Given that mABD = 16°, find mABC. 2. Given that mABD = (2x + 12)° and mCBD = (6x – 18)°, find mABC. Use the diagram for Items 3–4. 3. Given that FH is the perpendicular bisector of EG, EF = 4y – 3, and FG = 6y – 37, find FG. 4. Given that EF = 10.6, EH = 4.3, and FG = 10.6, find EG.

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Lesson Quiz: Part II 5. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints X(7, 9) and Y(–3, 5) . Then, put the equation in slope-intercept form.

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LM Bisects RT. Mark the congruent segments. T L M R

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