2

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.

3

Guided Practice: continued 2. Connect the midpoints to form the

midsegments , , and .

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.

8

Guided Practice: continued

The slope of

is

From point Y, draw a line that has a slope of

9

Guided Practice: continued

The slope of is 7

From point X, draw a line that has a slope of 7

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

Example

Does D lie on the perpendicular bisector of ? ?WhyAB

A

C

B

D

67

Example Is a perpendicular bisector of ? Why, or why not?CD

23333333333333 3AB

A

C

B

D

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

WARM-UP

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

ExamplesDoes the information given in the diagram allow you to conclude that P is on the angle bisector of angle A?

P

6

P

P6