Intro to Similarity Unit#5

Power Word definitions Ratio- a comparison between 2 Proportion – a statement in which two ratios are equal Scale Factor – The ratio of any two corresponding lengths in two similar geometric figures. Similar figures – figures whose corresponding sides have lengths that are proportional and whose corresponding angles are congruent.

Theorem: In a proportion, the product of the means equals the product of the extremes. 2 ways to write a proportion: 1) 2) Write a proportion using: 25, 45, 5, 9 Solve for b. b+ 5 : 7 = 5 :10 𝑏+5 7 = 5 10 25:5=45:9 10(b+ 5) = 7(5) 10b+50=35 10b = -15 b= -1.5 225=225

Solve for x.

What makes 2 shapes similar? Corresponding angles are congruent! Corresponding sides are in proportion! 4:2 = 2:1

𝐴𝐵 𝑋𝑌 𝐴𝐶 𝑋𝑍 𝐵𝐶 𝑌𝑍

What is the scale factor or ratio of similitude of the quadrilaterals? 8 4 = 4 2 = 6 3 So ratio of similitude is 2:1! 2 1 = 5 𝑁𝑀 2(NM)=5 NM=2.5

What evidence suggests the 2 triangles are similar? Parallel lines cut by a transversal form alternate interior angles that are congruent. Right angles are formed and congruent. If 2 corresponding pairs of angles in a triangle are congruent, then the 3rd pair is as well!

x 150 450 Label the diagram 100 𝑥 450 = 150 100 100x=67,500 X=675 cm

Scale drawings/maps Legs: 4:6, so 2:3 but bases are 1:1 Finding actual distance Examples: Maps: What is the actual distance between Hagerstown and Annapolis? Identify the scale factor 2) Legs: 4:6, so 2:3 but bases are 1:1 All sides must be in the same scale factor, so these are not similar trapezoids! About 4, so 4(24) =96 miles

Alex wants to reduce a figure that is 9 inches tall and 16 inches wide so that it will fit on a 9-inch-by-12-inch piece of paper. If he reduces the figure proportionally, what is the maximum size the reduced figure could measure? 15(4.5)=67.5 mm 30(4.5) = 136.35 mm 16 9 12 16 12 = 9 𝑥 So, 16x =108 x=6.75

𝑥+2 6 = 15 𝑥+1 x=-11 and x=8 10 6 = 20 𝐷𝐹 DF=12 (x+2)(x+1)=90 Algebraically speaking… 10 𝑥+2 6 = 15 𝑥+1 (x+2)(x+1)=90 (x+11)(x-8)=0 𝑥 2 +3x+2=90 𝑥 2 +3x-88=0 x=-11 and x=8 Factor! Substitute 8… 10 6 = 20 𝐷𝐹 DF=12

78𝑚 90𝑚 = 𝑥𝑐𝑚 36𝑐𝑚 𝑥=31.2 cm 4(4x+3) =60 16x+12=60 16x=48 X=3 1) Solve for x. 4(4x+3) =60 16x+12=60 16x=48 X=3 2) 78𝑚 90𝑚 = 𝑥𝑐𝑚 36𝑐𝑚 𝑥=31.2 cm

3) A Barbie doll is made to have a ratio of similitude of roughly 1:6 with a real 5’6” human being. If a Barbie doll has a waist size of 2.5 inches, how would you compute her “real-life” waist size? Hmm…Barbie is not very well modeled after a real person! 1 6 = 2.5 𝑥 𝑥=15 𝑖𝑛𝑐ℎ𝑒𝑠