Similarity of Triangles Click one of the buttons below or press the enter key BACK NEXT EXIT

In geometry, two polygons are similar when one is a replica (scale model) of the other. BACK NEXT EXIT

Consider Dr. Evil and Mini Me from Mike Meyers’ hit movie Austin Powers. Mini Me is supposed to be an exact replica of Dr. Evil. BACK NEXT EXIT

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The following are similar figures. II BACK NEXT EXIT

The following are non-similar figures. II BACK NEXT EXIT

Feefee the mother cat, lost her daughters, would you please help her to find her daughters. Her daughters have the similar footprint with their mother. Feefee’s footprint BACK NEXT EXIT

Which of the following is similar to the above triangle? 1. Which of the following is similar to the above triangle? B A C BACK NEXT EXIT

Note: One triangle is a scale model of the other triangle. BACK NEXT EXIT

How do we know if two triangles are similar or proportional? BACK NEXT EXIT

Triangles are similar (~) if corresponding angles are equal and the ratios of the lengths of corresponding sides are equal. BACK NEXT EXIT

The sum of the measure of the angles of a triangle is 1800. Interior Angles of Triangles A B C The sum of the measure of the angles of a triangle is 1800. Ð A + Ð B + ÐC =1800 BACK NEXT EXIT

Determine whether the pair of triangles is similar. Justify your answer. Answer: Since the corresponding angles have equal measures, the triangles are similar. Example 6-1b

If the product of the extremes equals the product of the means then a proportion exists. BACK NEXT EXIT

This tells us that  ABC and  XYZ are similar and proportional.   BACK NEXT EXIT

Q: Can these triangles be similar? BACK NEXT EXIT

Answer—Yes, right triangles can also be similar but use the criteria. BACK NEXT EXIT

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Do we have equality? This tells us our triangles are not similar. You can’t have two different scaling factors! BACK NEXT EXIT

If we are given that two triangles are similar or proportional what can we determine about the triangles? BACK NEXT EXIT

The two triangles below are known to be similar, determine the missing value X. BACK NEXT EXIT

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In the figure, the two triangles are similar. What are c and d ? B C P Q R 10 6 c 5 4 d BACK NEXT EXIT

In the figure, the two triangles are similar. What are c and d ? B C P Q R 10 6 c 5 4 d BACK NEXT EXIT

Sometimes we need to measure a distance indirectly Sometimes we need to measure a distance indirectly. A common method of indirect measurement is the use of similar triangles. h 6 17 102 BACK NEXT EXIT

MODELING A REAL-LIFE PROBLEM Error Analysis GEOMETRY CONNECTION Two students are visiting the mysterious statues on Easter Island in the South Pacific. To find the heights of two statues that are too tall to measure, they tried a technique involving proportions. They measured the shadow lengths of the statues at 2:00 P.M. and again at 3:00 P.M. 3:00 2:00

a b a b 2:00 3:00 a = b a = b a = b a = b a = b a = b Error Analysis SOLUTION They let a and b represent the heights of the two statues. Because the ratios of corresponding sides of similar triangles are equal, the students wrote the following two equations. 27 a 18 b 30 a 20 b 2:00 3:00 a 27 = b 18 a 30 = b 20 a = 27 18 b a = 30 20 b 30 ft 27 ft 18 ft 20 ft a = 3 2 b a = 3 2 b

Draw Similar Rectangles ABCD and EFGH whose lengths and widths are 16 and 12 and 12 and 9 respectively.

12 16 9 12

Two triangles are called “similar” if their corresponding angles have the same measure.      

a A b B c C Two triangles are called “similar” if their corresponding angles have the same measure.  Ratios of corresponding sides are equal. C A  a c     b B a A b B c C = =

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree?

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? 5.5 x 2 7 = Mary’s height Tree’s height Mary’s shadow Tree’s shadow = x 5.5 2 7

5.5 x 2 7 = 7 ( 5.5 ) = 2 x 38.5 = 2 x x = 19.25 The height of the tree is 19.25 feet

Find the missing measures if the pair of triangles is similar. Corresponding sides of similar triangles are proportional. and Example 6-2b

Find the cross products. Divide each side by 4. Answer: The missing measure is 7.5. Example 6-2b

Find the missing measures if each pair of triangles is similar. a. Answer: The missing measures are 18 and 42. Example 6-2c

Find the missing measures if each pair of triangles is similar. b. Answer: The missing measure is 5.25. Example 6-2c

Shadows Richard is standing next to the General Sherman Giant Sequoia three in Sequoia National Park. The shadow of the tree is 22.5 meters, and Richard’s shadow is 53.6 centimeters. If Richard’s height is 2 meters, how tall is the tree? Since the length of the shadow of the tree and Richard’s height are given in meters, convert the length of Richard’s shadow to meters. Example 6-3a

Let the height of the tree. Simplify. Let the height of the tree. Richard’s shadow Tree’s shadow Richard’s height Tree’s height Cross products Answer: The tree is about 84 meters tall. Example 6-3a

Answer: The length of Trudie’s shadow is about 0.98 meter. Tourism Trudie is standing next to the Eiffel Tower in France. The height of the Eiffel Tower is 317 meters and casts a shadow of 155 meters. If Trudie’s height is 2 meters, how long is her shadow? Answer: The length of Trudie’s shadow is about 0.98 meter. Example 6-3b

Similarity of Triangles Click one of the buttons below or press the enter key BACK NEXT EXIT