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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?

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Midsegments of triangles

Midsegments of Triangles


Gsp activity theorem 5 1
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

5

10

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.


Example problem
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?

Patty Paper Activity Theorem 5-2


Theorem 5 2

Theorem 5-2


Theorem 5 3 converse of thm 5 2

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



Homework
Homework p.293)

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


Review

B p.293)

R

S

A

C

T

Review

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


Review answers

B p.293)

R

S

A

C

T

Review Answers


Distance

Distance


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.

Patty Paper Activity


Activity continued

  • Open GSP file entitled “THM 5-4” ( make sure the sides are fairly long.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)

Activity (continued)


Proof

  • Given: make sure the sides are fairly long.

  • Prove:

  • StatementsReasons

Proof


Theorem 5 4

Theorem 5-4


Theorem 5 5 converse of thm 5 4

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


Homework1

Homework


Gsp activity

GSP Activity


Gsp activity1

GSP Activity


5 3 triangle bisectors

5-3 Triangle Bisectors the point lies on the bisector of the angle.


Terms

  • Concurrent the point lies on the bisector of the angle.

    • Three or more lines intersect at one point

  • Point of concurrency

    • The point at which three or more lines intersect

  • Circumscribed about

  • Inscribed in

Terms


Definitions of terms

Definitions of Terms


Theorem 5 6 concurrency of perpendicular bisectors

  • Circumcenter perpendicular to a segment at its midpoint- 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 5-6- Concurrency of Perpendicular Bisectors


Theorem 5 7 concurrency of angle bisectors

  • Incenter perpendicular to a segment at its midpoint- 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

Theorem 5-7- Concurrency of Angle Bisectors


Gsp activity2

GSP Activity


Gsp activity3

GSP Activity


5 4 medians and altitudes

5-4 Medians and Altitudes perpendicular to a segment at its midpoint


What do these numbers represent? perpendicular to a segment at its midpoint


What do these numbers represent? perpendicular to a segment at its midpoint


Definitions of terms1

Definitions of Terms


Definitions of terms2

Definitions of Terms


Theorem 5 8 concurrency of medians

  • Centroid- point opposite sideof 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 5-8- Concurrency of Medians


Theorem 5 9 concurrency of altitudes

  • Orthocenter- opposite sidepoint of concurrency of altitudes in a triangle

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

Theorem 5-9 Concurrency of Altitudes


Summary
Summary opposite side


Homework2

Homework


Class work

  • p. 312-313 #8-13, 24-27 opposite side

  • p.304-305 #2,15-19,26-28

  • p.296 #1-4, 6-8, 12-15, 18-22

  • Proofs p.298 #32 and 34

Class work


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