Mathematics » High School: Geometry » Similarity, Right Triangles, & Trigonometry

Understand similarity in terms of similarity transformations

• 1. Verify experimentally the properties of dilations given by a center and a scale factor:
  

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  • A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

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  • The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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• 2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

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• 3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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Prove theorems involving similarity

• 4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

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• 5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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Define trigonometric ratios and solve problems involving right triangles

• 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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• 7. Explain and use the relationship between the sine and cosine of complementary angles.

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• 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.^★

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Apply trigonometry to general triangles

• 9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

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• 10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.

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• 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

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