Midsegments of Triangles

GSP Activity Theorem 5-1

Draw triangle ABC.

Find and construct the midpoints of segments AB and AC and label them M and N respectively.

Measure <B, <C, <AMN, and <ANM. What do you notice?

What does this tell you about segments MN and BC and how do you know this?

Measure the length of segments MN and BC and compare. Calculate BC/MN to make a comparison.

Change the size of the triangle. Does the ratio BC/MN change or stays the same?

Theorem 5-1

A segment that joins the midpoints of two sides of a triangle ◦is parallel to the third side.◦is half as long as the third side.

5

10

Example problem

Points D, E, and F are midpoints of the sides of the triangle shown below. What are the lengths of the sides of the triangle? DF=30, AC=50, and BC=40

Patty Paper Activity Theorem 5-2

Construct a line segment. Label the endpoints A and B. Fold the line segment so that the endpoints lie on top of

one another. Crease the fold. Mark the point where the crease intersects the line as point C and a point on the crease but not on segment AB as point D.

What do you notice about the lengths of segments AC and CB? (measure the segments if needed)

What do you notice about <ACD and <BCD? What can you say about line CD with respect to segment

AB? Find the distance from point A to point D. Find the distance

from point B to point D. What do you notice about these distances?

What can you say about the two triangles that were formed?

Theorem 5-2

If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

Theorem 5-3(converse of Thm 5-2)

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

Using the Perpendicular Bisector Theorem (Problem 1 on p.293)

Homework

p. 288-289 #1-25 odd, 31, 33, 38, and 40

ReviewComplete the table for the figure below.

Assume points R, S, and T are midpoints of the respective sides.

B

R S

A T CAB BC AC ST RT RS

a. 12 14 18b. 15 22 10c. 10 9 8

Review Answers

AB BC AC ST RT RS12 14 18 6 7 920 15 22 10 7.5 1110 18 16 5 9 8

B

R S

A T C

Distance

The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line

Patty Paper Activity

Draw an angle. The angle can be obtuse, acute, or right, but make sure the sides are fairly long.

Make a fold through the vertex of the angle so that the two sides are on top of one another. Crease the fold. Draw a line along the crease. What is this new line called with respect to the angle?

Mark point A somewhere on the angle bisector Measure the distance from point A to each side of the

angle. Recall how distance is measured from a point to a line. You may need to make folds to ensure that the line is perpendicular to the sides of the triangle.

What do you notice about these distances? Form a conjecture about a point on the angle bisector and the distance to each side of the angle.

Activity (continued)

Open GSP file entitled “THM 5-4” (see wiki)Move point P around and observe the

relationship between EP and PF. Does your conjecture hold true for all locations of point P? If not, revise your conjecture.

What do you notice about the two triangles that are created?

What theorem or postulate confirms this? (SSS, ASA, SAS, AAS, or HL)

Proof

Given:

Prove:

Statements Reasons

Theorem 5-4

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

Theorem 5-5(converse of Thm 5-4)

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

Homework

5-2 Practice worksheet #1-19 all, 21-29 odd

GSP Activity

Perpendicular bisectors of a triangle

GSP Activity

Angle bisectors of a triangle

5-3 Triangle Bisectors

Terms

Concurrent◦Three or more lines intersect at one point

Point of concurrency◦The point at which three or more lines intersect

Circumscribed aboutInscribed in

Perpendicular Bisector- a line, ray , or segment that is perpendicular to a segment at its midpoint

Definitions of Terms

Theorem 5-6- Concurrency of Perpendicular Bisectors

Circumcenter- point of concurrency of the perpendicular bisectors of a triangle

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

The circumcenter is the point that is the center of the circle that contains each vertex of the triangle (circle is circumscribed about the triangle)

Theorem 5-7- Concurrency of Angle Bisectors

Incenter- point of concurrency of angle bisectors in a triangle

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

The incenter is the center of the circle that is inscribed in the triangle

GSP Activity

Medians in a Triangle

GSP Activity

Altitudes in a Triangle

5-4 Medians and Altitudes

What do these numbers represent?

What do these numbers represent?

Altitude- segment from a vertex that is perpendicular to the line that contains the opposite side

“altitude”- height or elevation (used in aviation, surveying, etc.); height above sea level of a location

Definitions of Terms

Definitions of Terms

Median- segment from a vertex to the midpoint of the opposite side

“median”- middle; divides in half

Theorem 5-8- Concurrency of Medians

Centroid- point of concurrency of medians in a triangle

The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.

Theorem 5-9 Concurrency of Altitudes

Orthocenter- point of concurrency of altitudes in a triangle

The lines that contain the altitudes of a triangle are concurrent.

Summary

Concurrent Lines

Perpendicular Bisectors

Angle Bisector

sMedians Altitudes

Name of Point of

ConcurrencyCircumcenter Incenter Centroid Orthocenter

Homework

5-3 Practice (12-23 all)5-4 Practice (1-11 all)

Class work

p. 312-313 #8-13, 24-27p.304-305 #2,15-19,26-28p.296 #1-4, 6-8, 12-15, 18-22Proofs p.298 #32 and 34