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Maria Elisa Vanegas 9-5

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A ratio is a comparison of 2 things it could be 2 values. Examples 1.A(-2,-1) B(4,3) rise 3-(-1) 4 2 run = 4-(-2) = 6 = 3 2. A(-1,3) B(1,4) rise 3-4 -1 1 run = -1-1 = -2 = 2 3. A(-2,-2) B(2,2) rise -2-2 -4 1 run = -2-2 = -4 = A proportion is simply a equation that tells us that 2 ratios are equal to each other. You solve proportions by cross multiplying the given fractions and then simplifying. You can check by inserting the variable to the equation and verifying. Examples 1.5 45 y = 63 5(63)=y(45) 315=45y y=7 2. x+2 2 4 6 = x+2 (x+2)²=6(24) (x+2)²=144 x+2= +/- 12 x+2=+/- 12 x= 10 or -14 3. 16 x-1 x-1 = 4 16(4)=x²-2 64=x²-2 ∫66=∫x² ∫66=x These 2 are related because they both involve ratios.

3 Polygons are similar iff they have corresponding angles that are congruent and their corresponding side lengths are proportional. Examples 1-3 Determine weather the polygons are similar. If so, write the similarity ratio and a similarity statement. 1. 2.

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3. EH = 30 = 2 EF = 90 = 2 AD 45 3 AB 135 3 135 45 90 30 A B D C E F H G

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The only thing that these does is that it helps determine how much something is enlarged or reduced. Examples 1. Multiply the vertices of the photo A B C D by 3/2. B (0,4) A (0,0) C (3,4) D (3,0) A(0,0) A(0 [3/2], 0[3/2]) A(0,0) B(0,4) B(0[3/2], 4[3/2]) B(0,6) C(3,4) C(3[3/2], 4[3/2]) C(4.5,6) D(3,0) C(3[3/2],0[3/2]) D(4.5,0) A(0,0) B(0,6) C(4.5,6) D(4.5,0)

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2. A(0,0) A(0[1/2], 0[1/2]) A(0,0) B(0,6) A(0[1/2], 6[1/2]) B(0,3) C(4.5,6) C(4.5[1/2], 6[1/2]) C(2.25,3) D(4.5,0) D(4.5[1/2], 0[1/2]) D(2.25,0) A(0,0) B(0,6) C(4.5,6) D(4.5,0) A(0,0) B(0,3)C(2.25,3) D(2.25,0) Multiply the vertices of the photo A B C D by 1/2.

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3. Multiply the vertices of the photo A B C D by 4/3. ROUND IF NEEDED A(0,0) A(0[4/3], 0[4/3]) A(0,0) B(0,8) B(0[4/3], 8[4/3]) B(0,11) C(4,8) C(4[4/3], 8[4/3]) C(5.3,11) D(4,0) D(4[4/3], 0[4/3]) D(5.3,0) A(0,0) B(0,8) C(4,8) D(4,0) B(0,11)C(5.3,11) D(5.3,0)

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o Right Triangle Similarity if you draw an altitude from the vertex of the right angle of a right triangle, you form 3 similar right triangles. o You do this by using ratios like shortest side/longest side of 2 similar triangles then you simplify. o This is an important skill because if someday you want to cut a tree of your house you have got to know how long it is so it doesn't crushes you house. x y 8 z 3 Examples Find all of the sides 1.x = 3 1.125 = y 3.2 = 3 3 8 y 9.125 1.125 z 8x=9 ∫y² = ∫10.27 3.375 = 3.2z x= 1.125 y=3.2 3.2 3.2 z=1.1

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6 ft 30 ft 3. Find the height of the tower. 6 = 30 30= x 6x = 900 6 X = 150 150 + 6 = Height= 156 ft x 2. Find the height of the Ceiba. 8 ft x 45 ft 8 = 45 45= x 2025= 8x 8 253.125= x 253.125+8= height =261.125 ft

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o Area- first you have to simplify the fraction of both shapes after you have done that you square the fraction. o Perimeter- first you find the perimeter of each shape with that you create a fraction of each perimeters then simplify. 6 4 3 12 1 7 24 14 6(4)=24 16 = 2 4(4)=16 24 3 3(2)+12(2)=30 1(2)+7(2)=16 16 8 30 = 15 14(4)= 56 24(24)=96 56= 7 96 12 1. Sides 40&25 40/25 = 8/5 (8/5)² = 64/25 2. Sides 30&12 30/12=5/2 (5/2) ²= 25/4 3. Sides 94&86 94/86=47/43 (47/43) ²= 2209/1849

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o Trigonometric= the study of triangles o Sin A= Opposite/Hypotenuse o Cos A= Adjacent/Hypotenuse o Tan A= Opposite/Adjacent o Solving a triangle means finding all of the angles and all of the sides. o These are useful to solve a right triangle because it helps you find the angles and the sides. Examples Write the ratio as a # and decimal rounded. R S T 13 12 5 Sin R= 12/13 ≈ 0.92 Cos T= 5/13 ≈ 0.38 Tan S= 5/12 ≈ 0.42

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100 m 40 ⁰ Tan 40 = x__ 100 100 (Tan 40) = x 83.90 x B 42 ⁰ x 12 Sin 42 = x/12 12(Sin 42)= x = 8.02 C A B 24 25 7 Cos A= 24/25 ≈ 0.96 Tan B= 24/7 ≈ 3.42 Sin B= 24725 ≈ 0.96 x y z12.6 cm 38 ⁰ Cos 38= 12.6/YZ YZ= 12.6/Cos 38 YZ= 15.99 cm

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o Angle of Elevation is a straight line going horizontally and another line that’s ABOVE the horizontal pointing somewhere, which together form the angle. o Angle of Depression is a straight line going horizontally and another line that’s BELLOW the horizontal pointing somewhere, which together form the angle. Angle of Depression Angle of Elevation

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Clasify each angle as angle of depression or elevation ball <1 <2 <3 <4 1.<1 is angle of elevation 2.<2 is anlge of depression 3.<3 is angle of elevation 4.<4 is angle of depression 5. P A x 41 ⁰ Tan 41= 4000/x x= 4000/Tan 41 x≈4601 ft 6. T S F x 7⁰7⁰ 90 ft Tan 7= 90/x x=90/Tan 7 x≈ 733 ft

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Introduction Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design at the corner,

Introduction Recognizing and using congruent and similar shapes can make calculations and design work easier. For instance, in the design at the corner,

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