5-6 to 5-7 Using Similar Figures What You’ll Learn To identify similar figures To find missing lengths in similar figures To use proportions to solve problems involving scale

Similarity Similar figures are figures that have the same shape, but not necessarily the same size. Imagine that you have reduced or enlarged a figure in a photocopy machine - the figure has the same shape, but not the same size. Similar triangles have the same angles, but the length of the sides are shorter or longer. However, the length of the sides must be proportional. This means that if one of the sides is twice as long as the corresponding side of the other triangle, then all the sides must be twice as long as the corresponding sides of the other triangle.

Similar Figures When two figures have the same shape, but not necessarily the same size they are similar. Corresponding angles Corresponding sides since 40/60=50/75=34/51 34 83 40 51 83 60 53 44 50 53 44 75

Similar Figures In similar triangles, corresponding angles have the same measure. Since 40/60 = 50/75 = 34/51, the corresponding sides are proportional You write ABC ~ FGH The symbol ~ means “is similar to”

Similar Polygons Two polygons are similar if: Corresponding angles have the same measure, and The lengths of the corresponding sides form equal ratios

Example 1: Verifying Similar Figures Verify that the triangles are similar Measures of X and P are 76.5 Measures of Y and Q are 41.5 Measures of Z and R are 62 Corresponding side form equal ratios XYZ ~ PQR Practice Text page 269 # 1-2

Activity Get with a partner Get a text book Turn to page 269 Work on numbers 1 and 2 Be prepared to explain you reasoning

Finding Missing Measures Example 2: If we have a triangle with side lengths of 2, 3, and 4 and a larger similar triangle with the shortest side equal to 6, what is the length of the other two sides of the triangle?

Solution: First, draw the figures so you can visualize the problem. Since the shortest side of the first triangle is 2, we know that 2 x 3 is 6, so the other sides are 3 times the sides of the first triangle. The other two sides are 9 and 12 (3 x 3 = 9 and 4 x 3 = 12).

Example 3: A rectangle has width equal to 10 feet and length equal to 16 feet. A similar rectangle has length equal to 22 feet. What is its width of the second rectangle?

First, draw the figures so you can visualize the problem

Step 1: Set up the proportion to solve for y. Step 2: Cross multiply. Step 3: Divide each side of the equation by 16 Practice: Text page 269 # 3-12 Assign Pr 5-6 Re 5-4 to 5-6 Quiz Test

Problem Solving A 6 ft. person is standing next to a flag pole has a shadow 4.5 ft long. The flag pole has a shadow of 15 ft long . What is the height of the flagpole? Draw a diagram Write a proportion Write cross products Simplify

Example 4 The following figures are similar. What is the value of x

Did I get Ya??? Note that in this problem, the given sides of the right triangles do not all correspond. The sides of lengths 5 and 10 correspond because they are the shorter legs of each of the given right triangles, but the side of length x is the longer leg of the triangle on the left and the side of length 13 is the hypotenuse of the triangle on the right. The Pythagorean Theorem needs to be applied to solve for the unknown longer leg.

The Pythagorean Theorem where a and b are the legs of the right triangle and c is the hypotenuse. Note that it does not matter if a is considered to be the shorter or longer leg

Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sums of the squares of the lengths of the legs

Label the triangle Hypotenuse Leg Leg

In ABC, B is a right triangle 122 + 52= 144 + 25 = 169 C2 = 169 C = square root 169 C = 13 x 12 B C 5

Example 4 The following figures are similar. What is the value of x

Find the missing sides 14 Geometry text page 538 7-15 8 16 5

Example 4: Applying the Pythagorean Theorem Step 1: Apply the Pythagorean Theorem to find the longer leg. Substitute 5 in place of a and 13 in place of c. Step 2: Simplify the terms in the equation. Step 3: Subtract 25 from both sides of the equation. Step 4: Take the square root of both sides and simplify.

Example 5: Applying the Pythagorean Theorem Step 5: Use b = 12 because a side of a triangle must be positive and label the diagram with the new information. Step 6: Determine the ratios of corresponding sides and set them equal to each other to create a proportion to solve for x. Each ratio in this proportion compares corresponding segments from the larger triangle on the left with the smaller triangle on the right. The ratio on the left compares the two shorter legs and the ratio on the right compares the two longer legs. Step 7: Solve the problem by cross multiplication. Multiply the numerator of the left fraction by the denominator of the right fraction to get 5x and then multiply the numerator of the right fraction by the denominator of the left fraction to get (12)(10) = 120. Solve by dividing both sides of the equation by 5 to get x = 24.

Work for Example 5 Geometry page 539 25-30

Scale Drawing A scale drawing represents an object's actual proportions, but in a smaller size. The scale is a ratio that compares the measurement on a map or drawing to the actual measurement. Extension project

Example 1: The scale of this drawing is: 2 centimeters equal 5 meters. What is the width of the bedroom?

Solution: One method is to use the ratio of the scale to determine the unknown length. We can use the ratio to write a proportion, using a variable to represent the width of the bedroom. From the scale drawing, we know the scale width of the bedroom is 6 cm.

Steps to this problem Step 1: Write the appropriate proportion. Let w represent the width of the bedroom. Step 2: Write the cross products. Multiply w by 2 and multiply 5 by 6. Step 3: Rewrite the equation with the new values. Step 4: Divide both sides of the equation by 2 to isolate the w. Step 5: 30 ÷ 2 = 15

How to work this problem out

Your turn… Your are to design a scale drawing of this room You may work with a partner, but each of you must turn in a project Your drawing should include: All dimensions of the room Actual measurements (in feet) Scale measurements (in inches) Ratios/Proportions to show that your drawing is done to scale Drawing will be worth 100 points total