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Guided Practice: The midpoints of a triangle are X (–2,
5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.
1. Plot the midpoints on a coordinate plane.
4
Guided Practice: continued 3. Calculate the slope of each
midsegment.
Calculate the slope of .
The slope of is
Slope formula
Substitute (–2, 5) and (3, 1) for (x1, y1) and (x2, y2).
Simplify.
5
Guided Practice: continued
Calculate the slope of .
The slope of is 7.
Slope formula
Substitute (3, 1) and (4, 8)
Simplify.
6
Guided Practice: continued
Calculate the slope of .
The slope of is
Slope formula
Substitute (–2, 5) and (4, 8)
Simplify.
7
Guided Practice:
4. Draw the lines that contain the midpoints.
The endpoints of each midsegment are the midpoints of the larger triangle.
Each midsegment is also parallel to the opposite side.
10Guided Practice: continued
The slope of is
From point Z, draw a line that has a slope of
The intersections of thelines form the vertices of the triangle.
Properties of Triangles Perpendicular and Angle
Bisectors
Objective: To use properties of perpendicular bisectors and angle bisectors
Perpendicular BisectorPerpendicular Bisector – a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
A
C
B
M
If is a perpendicular bisector of AB, then AP PB and AB
CPCP
23333333333333 3
23333333333333 3
P
EquidistantEquidistant from two points means that the distance from each point is the same.
A
C
B
C is equidistant from A and B;
therefore, CA = CB
Perpendicular Bisector Theorem
Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
A
C
B
DIf is the perpendicular bisector of AB,
then CA CB. Likewise, DA DB.
CD
23333333333333 3
P
AB CD 23333333333333 3
Converse of the Perpendicular Bisector
TheoremConverse of the Perpendicular Bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.
A
C
B
D
If DA DB, then D lies on the perpendicular bisector of AB.
P
Distance from a point to a line
The shortest distance from one point to another is a straight line.
Q
P
m
The distance from point P to line m is the length of QP
Examples
Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?
A
A
B
BC
C
A
B
CP
P
D
Angle Bisector TheoremAngle Bisector Theorem – If a point (D) is on the bisector of an angle, then it is equidistant from the two sides of the angle.
A
C
B
DIf , is the angle bisector of BAC,
then BD CD
AD
33333333333333
BD AB and DC AC 3333333333333333333333333333
Converse of the Angle Bisector Theorem
Converse of the Angle Bisector Theorem – If a point is on the interior of an angle, and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
A
C
B
D
If DB DC,
then AD lies on the bisector of BAC
33333333333333