S IMILAR S HAPES AND P ROPORTIONS MATH 8 MS. STEWART Outcome: E3 make and apply generalizations about the properties of similar 2-D shapes

H OW ABOUT A CLIP TO GET US STARTED ?

P ROPORTIONS What are proportions? - If two ratios are equal, they form a proportion. Proportions can be used in geometry when working with similar figures. What do we mean by similar? - Similar describes things which have the same shape but are not the same size = 1:3 = 3:9

E XAMPLES 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. 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. A proportion can be made relating the height and the width of the smaller figure to the larger figure: 2 feet 4 feet 8 feet 4 feet 4 2 = 8 4

S OLVING P ROPORTIONAL P ROBLEMS So how do we use proportions and similar figures? Using the previous example we can show how to solve for an unknown dimension. 2 feet 4 feet 8 feet ? feet

S OLVING P ROPORTION P ROBLEMS First, designate the unknown side as x. Then, set up an equation using proportions. What does the numerator represent? What does the denominator represent? Then solve for x by cross multiplying: 2 feet 4 feet 8 feet ? feet 4 2 = 8 x 4x = 16 X = 4

T RY O NE Y OURSELF Knowing these two stick figures are similar to each other, what is the ratio between the smaller figure to the larger figure? Set up a proportion. What is the width of the larger stick figure? 4 feet 8 feet 12 feet x feet

S IMILAR S HAPES 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.

P ROPORTIONS AND T RIANGLES What are the unknown values on these triangles? 16 m 20 m 4 m 3 m x y First, write proportions relating the two triangles = 3 x 4 = y 20 Solve for the unknown by cross multiplying. 4x = 48 x = 12 16y = 80 y = 5

T RIANGLES IN THE R EAL W ORLD Do you know how tall your school building is? There is an easy way to find out using right triangles. 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.

S OLVING FOR THE B UILDING ’ S H EIGHT Here is a sample calculation for the height of a building: 48 feet 4 feet 3 feet yardstick building x x 3 = x = 144 x = 36 The height of the building is 36 feet.