Proportions and Similar Figures


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  • Due to the math it does not make a difference whether the smaller side is the numerator or denominator. The only thing which matters is that it is consistent on both sides of the equation.
  • Knowing the two figures are similar the proportion between the two stick figures is 8 feet:12 feet. Once written as a fraction 8/12 reduces to 2/3. So the proportion between the two stick figures is 2:3 . If the proportion is 2:3 then the student should set up this equation and solve for x: 2 / 3 = 4 / x 2 * x = 3 * 4 x = 12 / 2 x = 6 feet
  • The right angles are equal, and the angles the shadow makes with the ground can assumed to be equal. They can be assumed to be equal because for objects close in distance the sun is the same angle from the ground. Thus the shadows have similar angles, so the triangles are similar. Also the hypotenuses do not matter in these triangles. You could solve for them using Pathagorean’s Theorem, but it isn’t required to solve the problem so we will leave them alone.
  • Proportions and Similar Figures

    1. 1. Proportions
    2. 2. Proportion <ul><li>Proportion is a statement that says two ratios are equal . </li></ul><ul><ul><li>In an election, Damon got three votes for each two votes that Shannon got. Damon got 72 votes. How many votes did Shannon get? </li></ul></ul><ul><ul><li>Damon 3 = 72 so 3 x 24 = 72 </li></ul></ul><ul><ul><li>Shannon 2 n 2 x 24 48 </li></ul></ul><ul><ul><li>n = 48, so Shannon got 48 votes. </li></ul></ul>
    3. 3. Proportion <ul><li>Tires cost two for $75. How much will four tires cost? </li></ul><ul><ul><li># of tires 2 = 4 so 2 x 2 = 4 tires cost 75 n 75 x 2 $150 </li></ul></ul><ul><ul><li>n = 150, so four tires cost $150 </li></ul></ul>
    4. 4. Proportion <ul><li>One more way to solve proportions: </li></ul><ul><li>2 = 6 2 x n = 6 x 8 2 n = 48 8 n 2 2 </li></ul><ul><li>n = 24 </li></ul>
    5. 5. Proportion <ul><li>Now you try! </li></ul><ul><li>Three cans of soup costs $5. How much will 12 cans cost? </li></ul><ul><li># of cans 3 = 12 3 x 4 = 12 cans cost 5 n 5 x 4 20 dollars </li></ul><ul><li>n = 20, so 12 cans cost $20 </li></ul>
    6. 6. Proportions <ul><li>What are proportions? </li></ul><ul><li>- If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures. </li></ul><ul><li>What do we mean by similar? </li></ul><ul><li>- Similar describes things which have the same shape but are not the same size. </li></ul>1 2 4 8 = 1:3 = 3:9
    7. 7. Examples <ul><li>These two stick figures are similar. As you can see both are the same shape. However, the bigger stick figure’s dimensions are exactly twice the smaller. </li></ul><ul><li>So the ratio of the smaller figure to the larger figure is 1:2 (said “one to two”). This can also be written as a fraction of ½. </li></ul><ul><li>A proportion can be made relating the height and the width of the smaller figure to the larger figure: </li></ul>2 feet 4 feet 8 feet 4 feet 4 ft 2 ft = 8 ft 4 ft
    8. 8. Solving Proportional Problems <ul><li>So how do we use proportions and similar figures? </li></ul><ul><li>Using the previous example we can show how to solve for an unknown dimension. </li></ul>2 feet 4 feet 8 feet ? feet
    9. 9. Solving Proportion Problems <ul><li>First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent? </li></ul><ul><li>Then solve for x by cross multiplying: </li></ul>2 feet 4 feet 8 feet ? feet 4x = 16 X = 4 4 ft 2 ft = 8 ft x ft
    10. 10. Try One Yourself <ul><li>Knowing these two stick figures are similar to each other, what is the ratio between the smaller figure to the larger figure? </li></ul><ul><li>Set up a proportion. What is the width of the larger stick figure? </li></ul>4 feet 8 feet 12 feet x feet
    11. 11. Similar Shapes <ul><li>In geometry similar shapes are very important. This is because if we know the dimensions of one shape and one of the dimensions of another shape similar to it, we can figure out the unknown dimensions. </li></ul>
    12. 12. Triangle and Angle Review <ul><li>Today we will be working with right triangles. Recall that one of the angles in a right triangle equals 90 o . This angle is represented by a square in the corner. </li></ul><ul><li>To designate equal angles we will use the same symbol for both angles. </li></ul>90 o angle equal angles
    13. 13. Proportions and Triangles <ul><li>What are the unknown values on these triangles? </li></ul>16 m 20 m 4 m 3 m x m y m First, write proportions relating the two triangles. Solve for the unknown by cross multiplying. 4x = 48 x = 12 16y = 80 y = 5 4 m 16 m = 3 m x m 4 m 16 m = y m 20 m
    14. 14. Triangles in the Real World <ul><li>Do you know how tall your school building is? </li></ul><ul><li>There is an easy way to find out using right triangles. </li></ul>To do this create two similar triangles using the building, its shadow, a smaller object with a known height (like a yardstick), and its shadow. The two shadows can be measured, and you know the height of the yard stick. So you can set up similar triangles and solve for the height of the building.
    15. 15. Solving for the Building’s Height <ul><li>Here is a sample calculation for the height of a building: </li></ul>48 feet 4 feet 3 feet yardstick building x feet 4x = 144 x = 36 The height of the building is 36 feet. x ft 3 ft = 48 ft 4 ft
    16. 16. Accuracy and Error <ul><li>Do you think using proportions to calculate the height of the building is better or worse than actually measuring the height of the building? </li></ul><ul><li>Determine your height by the same technique used to determine the height of the building. Now measure your actual height and compare your answers. </li></ul><ul><li>Were they the same? Why would there be a difference? </li></ul>