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Midsegments of Triangles

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Midsegments of Triangles

- 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?

5

10

- 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.

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

- 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?

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

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

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

B

R

S

A

C

T

Complete the table for the figure below. Assume points R, S, and T are midpoints of the respective sides.

B

R

S

A

C

T

- 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

- 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.

- 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)

- Given:
- Prove:
- StatementsReasons

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

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

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

- Perpendicular bisectors of a triangle

- Angle bisectors of a triangle

5-3 Triangle Bisectors

- Concurrent
- Three or more lines intersect at one point

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

- Circumscribed about
- Inscribed in

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

- 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)

- 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

- Medians in a Triangle

- 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

- Median- segment from a vertex to the midpoint of the opposite side
- “median”- middle; divides in half

- 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.

- Orthocenter- point of concurrency of altitudes in a triangle
- The lines that contain the altitudes of a triangle are concurrent.

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

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