Properties of Triangles Chapter 5

5.4 Midsegment Theorem State Standards for Geometry 17. Prove theorems using coordinate geometry. Lesson Goals Identify the midsegments of a triangle. Discover the Midsegment theorem Use properties of midsegments of a triangle to solve problems. ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

definition Midsegment of a Triangle A segment that connects the midpoints of two sides of a triangle. A B C D E

Discovering the Midsegment Theorem

example

x y (–2, 8) C D (10, 4) E (6, 0) Q P (4, 6) (2, 4) example

x y (–2, 8) C D (10, 4) E (6, 0) Q P (4, 6) (2, 4) example

x y (–2, 8) C D (10, 4) E (6, 0) Q P (4, 6) (2, 4) example

theorem Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. A B C D E

example

A B C L K J 10 3 example

x y M (0, 0) P (c, 0) N (a, b) QR example

x y M (0, 0) P (c, 0) N (a, b) QR

Parallel lines have the same slope. example x y M (0, 0) P (c, 0) N (a, b) QR

x y example First, find the slope of each midsegment.

x y Using each slope, and the noncollinear point, draw the lines that contain the sides. example

x y The intersections of each pair of lines are the vertices. example

Midsegments can be used to create fractal patterns. A fractal is a geometric shape that is made of smaller copies of the same geometric shape. The word fractal is from the Latin word fractus; translated means “fragmented and irregular.”

Fractal patterns are studied to understand nature. Blood vessel and nerve networks Branches and leaves Stream and river flows example

Polymer Chemistry Weather Prediction Capacitor Design Pavement Distress Oil Recovery Surfaces for Heat Transfer Other uses of fractal patterns included: example Earthquake Prediction Filters Video Compression Crystallization Computer Graphics

Midsegments can be used to create fractal patterns. Begin by drawing in the midsegments of the triangle. Repeat drawing in the midsegments of each white triangle. example

Summary What is a midsegment? What does the midsegment theorem say? What is a fractal?

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