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Section: G.3.7

Prove theorems about triangles using similarity

• Prove that a line parallel to one side of a triangle partitions the other two sides proportionally.
• Prove the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.
• Prove the Pythagorean Theorem using triangle similarity.
• Prove and use slope criteria for parallel and perpendicular lines.

Students describe dilations in terms of center and scale factor and use these terms to describe properties of dilations (for example, as a result of a dilation, the image of a line segment that does not include the center of dilation is parallel to its pre-image, with a proportional length determined by the scale factor). As in unit G1, students work with concepts from grade 8, but now use constructions created by hand or with software rather than physical models and transparencies. They develop a more precise definition of similarity in terms of a dilation, drawing on their knowledge of proportional relationships developed in grades 6 and 7. Students also use the definition of similarity to show that two objects are similar and establish the AA criterion for triangle similarity. With this knowledge they then prove and use theorems about triangles, prove and use slope criteria for parallel and perpendicular lines, construct points that partition a line segment into a given ratio, and explore why all circles are similar.

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Tasks

1 Midpoints of Triangle Sides

2 Pythagorean Theorem

3 Slope Criterion for Perpendicular Lines

4 Equal Area Triangles on the Same Base II