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Section 2.6

Section 2.6. Problem Solving in Geometry. Objective 1. Solve problems using formulas for perimeter and area. Geometry.

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Section 2.6

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  1. Section 2.6 • Problem Solvingin Geometry

  2. Objective 1 • Solve problems using formulas for perimeter and area.

  3. Geometry • Geometry is about the space you live in and the shapes that surround you.For thousands of years, people have studied geometry in some form to obtain a better understanding of the world in which they live. • In this section, we will look at some basic geometric formulas for perimeter, area, and volume.

  4. Geometry (Cont) • Remember that perimeter is a linear measure, that is, a measure of length. It will be measured in units such as feet, miles, or meters. • Area is a measure of square units and it will be measured in units such as square feet, square miles, or square meters. How many floor tiles are in your kitchen? That’s an area measure.

  5. Geometry (Cont) • Volume is a measure of cubic units. It measures the number of cubic units in a three dimensional object or rather the amount of space occupied by the figure. A liter is a measure of volume. Cups and gallons are also measures of volume. The volume of a solid is the number of cubic units that can be contained in the solid.

  6. Perimeter and Area of Rectangle • Common Formulas for Perimeter and Area • Square Rectangle • s l • s w

  7. Area of Triangle and Trapezoid • Common Formulas for Area • Triangle Trapezoid • b • h h • b a

  8. Example • A rectangle has a width of 10 inches and a perimeter of 125 inches. Find the length. • 1. Let l represent the unknown length. • 2. Represent other unknowns in terms of l. • there are no other unknowns • 3. Write an equation that describes l the condition.

  9. Example (cont) • A rectangle has a width of 10 inches and a perimeter of 125 inches. Find the length. • 4. Solve the equation and answer the question. • The length of the rectangle is 52.5 inches. • 5. Check the solution in the original problem. • The perimeter of a rectangle that is10 inches by 52.5 inches is 125 inches.

  10. Objective 1: Example • 1. A sailboat has a triangular sail with anarea of 24 square feet and a base that is 4 feet long. Find the height of the sail.  • Use the formula where • and

  11. Objective 1: Example (cont) • The height of the sail is 12 ft. 24 sq. ft. 4 ft

  12. Objective 2 • Solve problems using formulas for a circle’s area and circumference.

  13. The Circle • Formulas for Circles • radius • area • circumference • The circumference of a circle is a measure of length. The circumference of a circle is the distance around it, or its perimeter. r

  14. Example • Find the area and circumference of a circle with a radius of 5 inches. • square inches square inches • Circumference • inches inches

  15. Objective 2: Examples • 2a. The diameter of a circular landing padfor helicopters is 40 feet. Find the area andcircumference of the landing pad. Expressanswers in terms of . Then round answers to the nearest squarefoot and foot, respectively. • Use the formulas for the area andcircumference of a circle. The radius is 20 ft.

  16. Objective 2: Examples (cont) • Area: Circumference: • The area is ft2 The circumferenceor approximately is ft or 1256ft2or 1257ft2 approximately 126 ft

  17. Objective 2: Examples (cont) • 2b. Which one of the following is the better buy: a large pizza with an 18-inch diameter for $20.00or a medium pizza with a 14-inch diameter for$14.00? • The radius of the large pizza is 9 inches, andthe radius of the medium pizza is 7 inches. • large pizza: • medium pizza:

  18. Objective 2: Examples (cont) • For each pizza, find the price per square inch by dividing the price by the area. • Price per square inch for the large pizza: • Price per square inch for the medium pizza: • The large pizza is the better buy.

  19. Objective 3 • Solve problems using formulas for volume.

  20. Volume of a Cube and a Rectangular Solid • Common Formulas for Volume • Cube Rectangular Solid

  21. Volume of a Cylinder and a Sphere • Common Formulas for Volume • Circular Cylinder Sphere

  22. Volume of a Cone • Cone

  23. Example • Find the volume of a cylinder that has a height of 4 feet and a radius of 5 feet.

  24. Objective 3: Examples • 3. A cylinder with a radius of 3 inches and a height of 5 inches has its height doubled. How many times greater is the volume of the larger cylinder than the volume of the smaller cylinder? • Smaller cylinder: in., in. • The volume of the smaller cylinder is

  25. Objective 3: Examples (cont) • The volume of the smaller cylinder is • Smaller cylinder: in., in. • The height is doubled, so: • Larger cylinder: in., in. • The volume of the smaller cylinder is

  26. Objective 3: Examples (cont) • The ratio of the volumes of the two cylinders is • Thus, the volume of the larger cylinder is 2 times the volume of the smaller cylinder.

  27. Objective 4 • Solve problems involving the angles of a triangle.

  28. The Angles of a Triangle • The sum of the measures of the three angles of any triangle is 180º.

  29. Example • One angle of a triangle is three times another. The measure of the third angle is 10º more than the smallest angle. Find the angles. • 1.Let x represent one of the quantities. • 2.Represent other quantities in terms of x.

  30. Example (cont) • 3.Write an equation in x that describes the conditions. • (Sum of the anglesof a triangle is 180.) • One angle of a triangle is three times another. The measure of the third angle is 10º more than the smallest angle. Find the angles.

  31. Example(cont) • 4.Solve the equation and answer the question. • The angles are 34º, 102º and 44º. • 5.Check the solution. • 102º is three times 34º, thus one angle is three times another. The measure of the third, 44º is 10º more than the smallest angle. The angles sum to 180º.

  32. Objective 4: Examples • 4. In a triangle, the measure of the first angle isthree times the measure of the second angle.The measure of the third angle is lessthan the second angle. What is the measure of each angle? • Let the measure of the first angle. • Let the measure of the second angle. • Let the measure of the third angle.

  33. Objective 4: Examples (cont) • The three angle measures are 120°, 40°, and 20°.

  34. Objective 5 • Solve problems involving complementary and supplementary angles.

  35. Special Angles • Complementary Angles: • Two angles whose sum is 90º. • Supplementary Angles: • Two angles whose sum is 180º.

  36. Complements and Supplements • Algebraic Expressions for Angle Measures • Measure of an angle: x • Measure of the angle’s complement: • Measure of the angle’s supplement:

  37. Example • Find the measure of an angle that is 12º less than its complement. • 1.Let x represent one of the quantities. • measure of the first angle • 2.Represent other quantities in terms of x. • measure of the complement angle • 3. Write an equation in x that describes the conditions. The angle

  38. Example (cont) • 4. Solve the equation and answer the question. • measure of the first angle • measure of the complement angle • The angles are 39º and 51º. • 5.Check the solution. 39º is 12º less than 51º and the angles sum to 90º

  39. Example • Which one of the following is a better buy: a large pizza with a 16 inch diameter for $12.00 or two small pizzas, each with a 10 inch diameter, for $12.00? • 1. Since the cost is the same, we must just determine which has greatest area – the one big pizza or the two small ones. • 2.Find the area of the large pizza. • The diameter is 16, so the radius is 8. • which is approximately 201 square inches.

  40. Example (cont) • 3. Find the area of one small pizza. • The diameter is 10, so the radius is 5. • which is approximately 78.5 sq in. • 4. Find the area of two small pizzas. • The area of two small pizzas is about • Thus, the large pizza is the better buy.

  41. Objective 5: Examples • 5. The measure of an angle is twice the measure of its complement. What is theangle’s measure? • 1. Let the measure of the angle. • 2. Let the measure of its complement. • 3. The angle’s measure is twice that of its complement, so the equation is

  42. Objective 5: Examples (cont) • 4. Solve this equation • The measure of the angle is 60°.5. The complement of the angle is and 60° is indeed twice 30°.

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