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