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5.5: Special Right Triangles and Areas of Regular Polygons

5.5: Special Right Triangles and Areas of Regular Polygons. Expectations: G1.2.4 : Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles.

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5.5: Special Right Triangles and Areas of Regular Polygons

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  1. 5.5: Special Right Triangles and Areas of Regular Polygons Expectations: G1.2.4: Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º triangles and 45º- 45º- 90º triangles. G1.5.1: Know and use subdivision or circumscription methods to find areas of polygons G1.5.2: Know, justify and use formulas for the perimeter and area of a regular n- gon. 5.5: Special Right Triangles

  2. ACT Prep • If one diagonal of a rhombus is 12 inches long and the other is 32 inches long, how many inches long, to the nearest hundredth of an inch, is a side of the rhombus? • 8.54 • 17.09 • 34.17 • 35.78 • 48.00 5.5: Special Right Triangles

  3. If a square has area of x2 square units, what is the length of one of its diagonals? 5.5: Special Right Triangles

  4. 45-45-90 Right Triangle Theorem • If a leg of a 45-45-90 right triangle is x units long, then the hypotenuse is x√2 units long. 5.5: Special Right Triangles

  5. 30-60-90 Right Triangles • a. sketch an equilateral triangle with sides of 2x units long. • b. draw an altitude of the triangle. • c. label all known measures. • d. what is the length of the altitude? 5.5: Special Right Triangles

  6. 30-60-90 Right Triangle Theorem In a 30-60-90 right triangle, if the length of the shorter leg is x units, then the longer leg is x√3units and the hypotenuse is 2x units long. 5.5: Special Right Triangles

  7. The hypotenuse of a 30-60-90 right triangle is 20 cm. What are the lengths of the other 2 sides? 5.5: Special Right Triangles

  8. What is the perimeter of a 30-60-90 right triangle if the length of the hypotenuse is 8 mm? 5.5: Special Right Triangles

  9. ACT Prep • If the length of a diagonal of a square is 18 inches long, what is the area of the square, in square inches? • 9√2 • 36√2 • 72 • 162 • 324 5.5: Special Right Triangles

  10. ACT Prep • If the length of each side of a regular hexagon is 10 centimeters long, what is the area of the hexagon, to the nearest centimeter? • 25√3 • 60 • 100√3 • 150√3 • 600√3 5.5: Special Right Triangles

  11. Center of a Regular Polygon • The center of a regular polygon is the point which is equidistant from the vertices of the regular polygon. 5.5: Special Right Triangles

  12. Apothem of a regular polygon • An apothem of a regular polygon is a segment with one endpoint at the center of the regular polygon and the other endpoint on the polygon, such that the segment is perpendicular to a side of the polygon. 5.5: Special Right Triangles

  13. Center and apothem of a regular polygon Center of the regular octagon Apothem of the regular octagon 5.5: Special Right Triangles

  14. Area of a Regular Polygon • Locate the center of the regular polygon. • Triangulate the polygon using the center as a common vertex. • What type of triangles are formed? • Draw the altitudes of the triangles. 5.5: Special Right Triangles

  15. Area of a Regular Polygon • 5.What are the altitudes in terms of the polygon? • 6. What is the area of one triangle? • 7. What is the area of the regular polygon expressed as a product? • 8. Change to using the perimeter. 5.5: Special Right Triangles

  16. Area of a Regular Polygon Theorem • If a regular polygon has area of A square units, perimeter of p units and an apothem of a units, then • A = 5.5: Special Right Triangles

  17. Assignment • pages 336-338, numbers 10-17(all), 22-38(evens), 44, 45 5.5: Special Right Triangles

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