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Geometry

9. Geometry. Ancient and Modern Mathematics Embrace. Perimeter and Area. 9.3. Calculate the perimeter and area of geometric objects. Understand how area formulas for geometric figures are related. Use the Pythagorean theorem to solve problems involving right triangles.

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Geometry

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  1. 9 Geometry Ancient and Modern Mathematics Embrace

  2. Perimeter and Area 9.3 • Calculate the perimeter and area of geometric objects. • Understand how area formulas for geometric figures are related. • Use the Pythagorean theorem to solve problems involving right triangles. • Calculate circumference and area of circles.

  3. Perimeter and Area

  4. Deriving Area Formulas

  5. Deriving Area Formulas

  6. Deriving Area Formulas

  7. Deriving Area Formulas • Example: A recreation area is in the shape of a parallelogram. If a pound of grass seed covers 100 square yards, how much grass seed is needed to seed the entire area? (continued on next slide)

  8. Deriving Area Formulas • Solution: 30,000 ÷ 100 = 300 pounds of seed

  9. Deriving Area Formulas • Example: If a pound of grass seed covers 100 square yards, how much grass seed is needed to seed only the triangular area ACD? (continued on next slide)

  10. Deriving Area Formulas • Solution: 5,250 ÷ 100 = 52.5 pounds of seed

  11. Deriving Area Formulas

  12. Deriving Area Formulas • Example: A gardener wants to fill a triangular flower bed (shown below) with flowers. The gardener estimates that he will need four bulbs for each square foot of the flower bed. How many dozen bulbs should he buy? (continued on next slide)

  13. Deriving Area Formulas • Solution: We will use Heron’s Formula.

  14. Deriving Area Formulas

  15. Deriving Area Formulas • Example: A platform for a statue will be formed by joining four congruent trapezoids and one square (shown below). Use the formulas we have developed so far to determine the surface area of the platform. (continued on next slide)

  16. Deriving Area Formulas • Solution: • Area of top: • Area of each side: • Total area:

  17. The Pythagorean Theorem

  18. The Pythagorean Theorem • Example: Use the lengths of the two given sides to find the length of the third side in the given triangle. • Solution: • Given: and • Pythagorean Theorem:

  19. The Pythagorean Theorem • Example: Find the length of the third side in the given triangle. • Solution: • Given: and • Pythagorean Theorem:

  20. The Pythagorean Theorem • Example: The Great Pyramid in Egypt has a square base measuring 230 meters on each side, and the distance from one corner of the base to the tip of the pyramid is 219 meters. What is the height of the pyramid? (continued on next slide)

  21. The Pythagorean Theorem • Solution: • Distance to center of base: • Height of pyramid:

  22. Circle The number π, which appears in these formulas, is the Greek letter pi. We will use 3.14 to approximate pi.

  23. Circles • Example: A man is designing a basketball court in the form of a segment of a circle. There will be a fence at the rounded end of the court, and he will paint the court with a concrete sealer. How much fencing is required? (continued on next slide)

  24. Circles • Solution: • The court is one-sixth of the interior of a circle with a radius of 20 feet. • Circumference of full circle: • One-sixth of the circumference: 20.93 feet

  25. Circles • Example: A man is designing a basketball court in the form of a segment of a circle. There will be a fence at the rounded end of the court, and he will paint the court with a concrete sealer. What is the area of the surface of the court? (continued on next slide)

  26. Circles • Solution: • The court is one-sixth of the interior of a circle with a radius of 20 feet. • Area of full circle: • One-sixth of the area: 209.3 square feet

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