Similar Figures, Scale Drawings, and Indirect Measure More ways to use a ratio!!!!

Our Objectives? We will use the skills practiced in solving proportions to find the missing value in similar figures. We will apply the concepts of ratios to scale. We will extend the idea of similar geometric shapes to find indirect measures. We will start by applying the process to geometric shapes.

Looking back….. Last year you discussed geometric shapes that were congruent and geometric shapes that were similar. Who remembers the difference? Similar shapes have the same shape, but they are not the same size. The size is changed proportionately. Congruent shapes have the same shape and same size. This is the basis for the beginning of this lesson….

Do you know what corresponding parts are??????? Similar figures have two properties: Corresponding sides are proportional Corresponding angles are congruent ABC ~ DEF Fill in information on your notes! D 53o 53o D L A corresponds to L ___ A 9 6 L C corresponds to L ___ F E 90o L B corresponds to L ___ 10 15 90o B E __ AC corresponds to ___ __ DF 8 12 C 37o __ DE 37o __ AB corresponds to ___ F __ BC corresponds to ___ __ EF

I will choose to start with side AC and its corresponding side, DF So….if the sides are proportional, then we should be able to write equal ratios from corresponding sides. I will choose to start with side AC and its corresponding side, DF Now I’ll go back to the small triangle and choose another side and its corresponding “partner.” D 53o 53o 90 = 90 A 9 6 10 15 = 6 9 90o 10 15 90o B E 8 12 C 37o 37o F Use cross products to test for a proportion. The sides of these similar triangles are proportional. Notice the movement between the similar figures……..

Proportions are used to find missing sides in similar figures. 24 16 n 18 Where you start doesn’t matter. The pattern is what matters! When setting up your proportion, think about a ping pong game between corresponding sides. Choose a starting place and ping-pong between the shapes to set up your ratios. n _ = 16 18 24 24 = 18 16 n 2 3 6 16(18) = 24 n _____ ____ 24 24 16(18) = n 24 24 12 = n

A Pentagon ABCDE ~ Pentagon FGHIJ 12 E B F 6 ?? J G 14 ? 7 I H D C 5 10 __ __ Find the length of sides JF and JI. 12 = 10 ?? 5 ? = 5 14 10 (5)12 = 10n ?? ____ _____ 10 10 ____ _____ 10 10 (5)14 = 10n 2 2 6 = ?? 7 = n

p 14 cm 8 cm 17 ½ cm n 10 cm 12 cm 21 cm The two pentagons are similar. This means their sides are proportional, and their angles are congruent. Let’s first find the length of side “n.” 7 4 2 1 7 4 5 2 p = 14 cm

Find the length of z. If angle A is 35o, what is the measure of angle R? 8 z 7 5 A R The measure of angle R would also be 350 5_ = z__ 7 8 ____ ____ 7 7 5(8) = 7z 5.71 = z (rounded)

This same process can be used to find indirect measurements This same process can be used to find indirect measurements. Indirect measures are measures found using the idea of similar figures and corresponding sides If we want to find the height of the tree, we can use the mailbox and shadow to find this, using a proportion. Imagine a triangle formed with each object. These triangles will be our similar shapes.

Choose the measures for your proportion Choose the measures for your proportion. We want to find the height of the tree. A nearby mailbox that is 3 ft. tall has a shadow that is 2’ long. The tree’s shadow is 5ft. Long. height. 3 ft. 5 _ = n_ 2 3 2 ft. 5 ft. Shadow is to shadow as height is to height _____ ______ 2 2 (5)3 = 2n 7 ½ ft = height of the tree

Measurement Application A rocket casts a shadow that is 91.5 feet long. A 4-foot model rocket casts a shadow that is 3 feet long. How tall is the rocket? Write a proportion using corresponding sides. h 4 __ 91.5 3 ____ = Write the equation and solve. 3h ____ _ 4(91.5) 3 3 = 122 = h The rocket is 122 feet tall.

The Fried Bird 5 = 3 h 21 (21) 5 = 3 h 35 feet = h On a sunny day, a telephone pole casts a shadow 21 ft long. A 5-foot-tall mailbox next to the pole casts a shadow 3 ft long. How tall is the pole? 5 = 3 h 21 7 1 (21) 5 = 3 h h 5ft 21 ft 3 ft. _____ _____ 3 3 35 feet = h

On a sunny afternoon, a goalpost casts a 75 ft shadow. A 6 On a sunny afternoon, a goalpost casts a 75 ft shadow. A 6.5 ft football player next to the goal post has a shadow 19.5 ft long. How tall is the goalpost? h = 75 6.5 19.5 75(6.5) = 19.5h ____ ____ 19.5 19.5 h 6.5 ft 75 ft. 19.5 ft. 25 feet = h

Using the same process, find the height of this tree…… n_ = 10_ 5 4 5(10) = 4 n _____ ______ 4 4 12 ½ ft. = height of the tree

Scale is another way to use proportions…. A scale drawing is an enlarged or reduced drawing that is similar to an actual object or place. The ratio of a scale (represented) distance to actual distance is the scale. The scale of a map is 1in. : 50 miles. A map of Tennessee would show a distance of 3 ½ inches from Nashville to Crossville. What is the actual distance between these two cities? Complete this proportion in your notes to solve. drawing actual 1 in_ = 3 ½ inches_ 50 miles n miles 50(3.5) = 1 n 175 = miles

In this scale drawing of someone’s house, 1 cm = 12 ft In this scale drawing of someone’s house, 1 cm = 12 ft. Find the actual dimensions of the living room and kitchen. 1.8 cm Kitchen 1.5 cm 𝟐𝟏.𝟔 𝒇𝒕. 1 𝑐𝑚 12 𝑓𝑡 = 1.8 𝑐𝑚 ??? = 1.8(12) 1 18 𝒇𝒕. 1 𝑐𝑚 12 𝑓𝑡 = 1.5 𝑐𝑚 ??? = 1.5(12) 1 Living Room 1.5 cm 2 cm 18 𝒇𝒕. = 1.5(12) 1 𝟐𝟒 𝒇𝒕. Drawing actual 1 cm_ = _____ cm_ 12 ft actual distance 1 𝑐𝑚 12 𝑓𝑡 = 1.5 𝑐𝑚 ??? Drawing actual 1 cm_ = _____ cm_ 12 ft actual distance 1 𝑐𝑚 12 𝑓𝑡 = 2 𝑐𝑚 ??? = 2(12) 1

Did we meet our objective of solving problems with proportions????