Geometry: Wednesday March 28th Today’s Agenda Take out writing assignment and put in Geometry Inbox folder Discussion of Box Project (4/3) Notes on Section 7.1, 7.2 & maybe (7.5) HW: Page 457: # 8 – 15 Page 465: # 1- 6 Do Now What is difference between congruent figures and similar figures? Today’s Objectives Understand similar figures and how to apply to real world problems

Similar Figures Susan Phillips Lee’s Summit, MO

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.

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 For each part of one similar figure there is a corresponding part on the other figure. Segment AB corresponds to segment DE. B A C E Name another pair of corresponding segments. D F

Similar Figures Angle A corresponds to angle D. B Name another pair of corresponding angles. A C E D F

Similar Figures Corresponding sides have lengths that are proportional. Corresponding angles are congruent.

Similar Figures W Z X Y 9 cm 6 cm A D B C 3 cm 2 cm Corresponding sides: AB corresponds to WX. BC corresponds to XY. CD corresponds to YZ. AD corresponds to WZ.

Similar Figures W Z X Y 9 cm 6 cm A D B C 3 cm 2 cm Corresponding angles: A corresponds to W. B corresponds to X. C corresponds to Y. D corresponds to Z.

Similar Figures W Z X Y 9 cm 6 cm A D B C 3 cm 2 cm In the rectangles above, one proportion is = , or = . AB WX AD WZ 2 6 3 9 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. 100 200 ____ 111 y ___ Write a proportion using corresponding side lengths. = 200 • 111 = 100 • y The cross products are equal.

The two triangles are similar. Find the missing length y. y is multiplied by 100. 22,200 100 ______ 100y 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. B A 60 m 120 m 65° 50 m 100 m 45° 52 m y 50 100 ____ 52 y ___ = Write a proportion using corresponding side lengths. 5,200 = 50y 5,200 50 _____ 50y 50 ___ Divide both sides by 50 to undo the multiplication. = 104 m = y

Try This The two triangles are similar. Find the missing length y and the measure of B. A B 60 m 120 m 50 m 100 m y 52 m 65° 45° 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? Reduced Actual 2 54 3 w

Using Proportions with Similar Figures Reduced Actual 2 54 3 w 3 cm w cm 2 cm 54 cm _____ = Write a proportion. 54 • 3 = 2 • w The cross products are equal. 162 = 2w w is multiplied by 2. Divide both sides by 2 to undo the multiplication. 81 = w

Try these 5 problems. These two triangles are similar. 1. Find the missing length x. 2. Find the measure of J. 3. Find the missing length y. 4. Find the measure of P. 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 to be 15 ft, what will the width be? 30 in. 36.9° 4 in. 90° 12 ft

In Algebra 1 you learned the Cross Products Property In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products.

Example : Solving Proportions Solve the proportion. 7(72) = x(56) Cross Products Property 504 = 56x Simplify. x = 9 Divide both sides by 56.

Example: Solving Proportions Solve the proportion. (z – 4)2 = 5(20) Cross Products Property (z – 4)2 = 100 Simplify. (z – 4) = 10 Find the square root of both sides. Rewrite as two eqns. (z – 4) = 10 or (z – 4) = –10 z = 14 or z = –6 Add 4 to both sides.

Check It Out! Example Solve the proportion. 3(56) = 8(x) Cross Products Property 168 = 8x Simplify. x = 21 Divide both sides by 8.

Check It Out! Example Solve the proportion. 2y(4y) = 9(8) Cross Products Property 8y2 = 72 Simplify. y2 = 9 Divide both sides by 8. y = 3 Find the square root of both sides. y = 3 or y = –3 Rewrite as two equations.

Check It Out! Example Solve the proportion. d(2) = 3(6) Cross Products Property 2d = 18 Simplify. d = 9 Divide both sides by 2.

Check It Out! Example Solve the proportion. (x + 3)2 = 4(9) Cross Products Property (x + 3)2 = 36 Simplify. (x + 3) = 6 Find the square root of both sides. (x + 3) = 6 or (x + 3) = –6 Rewrite as two eqns. x = 3 or x = –9 Subtract 3 from both sides.

Homework Page 457: # 8 – 15 Page 465: # 1- 6