5-1

equidistant: _______________________________________________________________

locus: ____________________________________________________________________

Theorem: Hypothesis: Conclusion:

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 the segment, then it is on the

perpendicular bisector of the segment.

begin with this conclude this

Ex. 1: Find each measure.

A. MN = _____ B. BC = 2 x _____ = _______

C. TU 3x + 9 =

= x

TU = ____________ = 3 (____) + 9 = ______ + 9 = _______

Theorem Hypothesis Conclusion

Angle Bisector Theorem: If a point is on the angle

bisector of an angle, then it 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 it is on

the bisector of the angle.

Ex. 2: Find each measure.

A. BC = _____ B. m EFH, given that m EFG = 50°

m EFH = ½ (m EFG)

= 25°

m EFH =

C. m MKL

3a + 20 =

a =

m MKL = _____________ = 2(___) + 26 =

__ __

Ex. 3: John wants to hang a spotlight along the back of a display case. Wires AD and CD are the

same length, and A and C are equidistant from B. How do the wires keep the spotlight

centered?

__ __ __

Since AD CD, _____ is on the bisector of AC.

(point)

B is the midpoint of ______ , so ______ is the perp. bisector of ______.

(segments in these blanks)

Thus D is centered below _____. ( point)

Ex. 4: Write an equation in point-slope form for the perp. bisector of the segment with endpoints

C (6, -5) and D (10, 1).

using the two points, find the slope first:

because the slope of this segment is _______, the slope of its perp.

bisector is ______ m = _____

next, using the two points, find the midpoint:

because this is the midpoint of the segment, it is where the perp.

bisector crosses – which means this point is a point on the line

(x1, y1) = (___, ___)

now, write the equation using point-slope form: y – y1 = m (x – x1)