Survey

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Similar Figures M.C. Escher Some artists use mathematics to help them design their creations. In M.C. Escher’s Square Limit, the fish are arranged so that there are no gaps or overlapping pieces. Square Limit by M.C. Escher How are the fish in the middle of the design and the surrounding fish alike? How are they different? Square Limit by M.C. Escher Escher used a pattern of squares and triangles to create Square Limit. These two triangles are similar. Similar figures have the same shape but not necessarily the same size. Similar Figures For each part of one similar figure there is a corresponding part on the other figure. Segment AB corresponds to segment DE. Name another pair of corresponding segments. B A C D E F Similar Figures Angle A corresponds to angle D. B Name another pair of corresponding angles. A C D E F Similar Figures •Corresponding sides have lengths that are proportional. • Corresponding angles are congruent. Congruent Figures In order to be congruent, two figures must be the same size and same shape. Similar Figures Similar figures must be the same shape, but their sizes may be different. Similar Figures This is the symbol that means “similar.” These figures are the same shape but different sizes. Similar Figures A 3 cm 2 cm B D 2 cm 3 cm W 9 cm 6 cm Z 6 cm C X Corresponding sides: AB corresponds to WX. BC corresponds to XY. CD corresponds to YZ. AD corresponds to WZ. 9 cm Y Similar Figures A 3 cm 2 cm B D 2 cm 3 cm W 9 cm 6 cm Z 6 cm C Corresponding angles: A corresponds to B corresponds to W. X. C corresponds to D corresponds to Y. Z. X 9 cm Y Similar Figures A 3 cm 2 cm B D 2 cm 3 cm W 9 cm 6 cm Z 6 cm C X 9 cm In the rectangles above, one proportion is AB AD 2 3 = , or = . WX WZ 6 9 Y If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar. Missing Measures in Similar Figures The two triangles are similar. Find the missing length y and the measure of D. 100 111 Write a proportion using ____ = ___ 200 y corresponding side lengths. 200 • 111 = 100 • y The cross products are equal. The two triangles are similar. Find the missing length y. 22,200 = 100y 22,200 100y ______ = ____ 100 100 y is multiplied by 100. Divide both sides by 100 to undo the multiplication. 222 mm = y The two triangles are similar. Find the measure of angle D. Angle D is congruent to angle C. If angle C = 70°, then angle D = 70° . Try This The two triangles are similar. Find the missing length y and the measure of B. A 60 m 65° 50 m 45° 52 m B 120 m 100 m y 50 52 ____ ___ = 100 y 5,200 = 50y 5,200 = 50y _____ ___ 50 50 104 m = y Write a proportion using corresponding side lengths. Divide both sides by 50 to undo the multiplication. Try This The two triangles are similar. Find the missing length y and the measure of B. A 60 m 65° 50 m 45° 52 m B 120 m 100 m y Angle B is congruent to angle A. If angle A = 65°, then angle B = 65° Using Proportions with Similar Figures This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting? Actual Reduced 2 54 3 w Using Proportions with Similar Figures Actual Reduced 2 54 3 w 2 cm 3 cm _____ = Write a proportion. w cm 54 cm 54 • 3 = 2 • w The cross products are equal. 162 = 2w 81 = w w is multiplied by 2. Divide both sides by 2 to undo the multiplication. Try these 5 problems. These two triangles are similar. 1. Find the missing length x. 30 in. 2. Find the measure of J. 36.9° 3. Find the missing length y. 4 in. 4. Find the measure of P. 90° 5. Susan is making a wood deck from plans for an 8 ft by 10 ft deck. However, she is going to increase its size proportionally. If the length is 12 ft to be 15 ft, what will the width be? Joan used a mirror to estimate the height of a flagpole. What is the height of the flagpole? 5 ft flagpole 2 ft---|-----------------9 ft----------------------- 2 5 9 x 2 x 95 x 22.5 ft 2 x 45