WARM UP: In Chapter 1, you learned the definition of

a midpoint of a segment. What do you think a midsegment of a triangle is?

Find the midpoint of AB:o A(-2, 3), B(4, 1)

o A(0, 5), B(3, 6)

5.1 and 5.2 Midsegments of Triangles and

Perpendicular and Angle Bisectors

I can use properties of midsegments to solve problems. I can use properties of perpendicular bisectors and angle bisectors.

There are two special relationships between a midsegment of a triangle and the third side of the

triangle.

Triangle Midsegments Theorem

Theorem If… Then…

If a ________________________________ of two sides of a

triangle, then the segment is ________________________________________________.

D is the midpoint of CA and E is the midpoint of

CB

Problem: Identifying Parallel Segments

What are the three pairs of parallel segments in triangle DEF?

Problem: Identifying Parallel Segments

In triangle XYZ, A is the midpoint of XY, B is the midpoint of YZ, and C is the midpoint of ZX. What are the three pairs of parallel segments?

Problem: Identifying Parallel Segments

What are the three pairs of parallel segments in triangle DEF?

Problem: Finding Lengths

In triangle QRS, T, U, and B are midpoints. What are the lengths of TU, UB, and QR?

Problem: Finding Length

In the figure, AD = 6 and DE = 7.5. What are the lengths of DC, AC, EF, and AB?

Problem: Finding Length

In triangle RST, G, H, and K are midpoints. If RS = 28, GH = 20, and RH = 22, what is the length of KH, ST, and GK?

Problem: Using a Midpoint of a Triangle

Complete the three application problems.

1. .

2. .

3.

In the diagram below on the left, CD is the perpendicular bisector of AB. CD is perpendicular

to AB at its midpoint. In the diagram below on the right, CA and CB are drawn to complete

triangle CAD and triangle CBD.

A point is _______________from two objects if it is the same distance from the objects. So point C is

_____________ from points A and B.

There is a special relationship between the points on the perpendicular bisector of a segment and

the endpoints of the segment.

Perpendicular Bisector Theorem

Theorem If… Then…

If a point is on the perpendicular bisector of a segment, then it is equidistance from the endpoints of the

segment.

Converse of the Perpendicular Bisector Theorem

Theorem If… Then…

If a point is equidistant from the

endpoints of a segment, then it is on

the perpendicular bisector of the

segment.

Problem: Using the Perpendicular Bisector

Theorem BD is the perpendicular bisector of AC, so B is

equidistant from A and C.

Problem: Using the Perpendicular Bisector

Theorem What is the length of QR?

Problem: Using the Perpendicular Bisector

Theorem What is MN?

Angle Bisector Theorem

Theorem If… Then…

If a point is on the bisector of an angle,

then the point is equidistant from the sides of the angle.

Converse of Angle Bisector Theorem

Theorem If… Then…

If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

Problem: Using the Angle Bisector Theorem

What is the length of RM?

Problem: Using the Angle Bisector Theorem

What is the length of FB?

Problem: Using the Angle Bisector Theorem

What is the length of OP?

After: Lesson Check1. Which segment is parallel to JK?2. If LK = 46, what is NM?3. If JK = 5x + 20 and NO = 20, what is the value of x?

4. What is the relationship between AC and BD?5. What is the length of AB?6. What is the length of DC?

Homework:Page 288, #8-26, 32, 34 and Page 296, #6-9 all, 12-22 all