abababb RATIO – a ratio compares two numbers by dividing. The ratio of two numbers can be written in various ways such as a to b, a:b, or a/b, where b doesn’t equal 0. For example the ratios of 3 to 4 can be represented as 3:4 or ¾ abcdad b c abcd PROPORTION – a proportion is an equation stating that two ratios are equal. In this proportion a/b = c/d, a and d are the extremes when b and c are the means. When a proportion is written as a:b = c:d, the extremes are in the first and last positions, which means that the means are in the two middle positions. Proportions and ratios are related since proportions use ratios so inevitably, they both compare two number by division and can be used to create detailed miniature models.

abcdbd adbc abcd To solve a proportion, you need to use the Cross Products Property (in a proportion, if a/b = c/d and b and d don’t = 0, then ad = bc) so according to the Properties of Proportions, the proportion a/b = c/d is equivalent to the following: adbc ad = bc badc b/a =d/c acbd a/c = b/d

EX1. 3:4 = 12:x 3x = 48 x=16 EX2. 5/17 = x/6 30 = 17x x = 1.76 EX to 2 = x to = 2x x = 148.5

Two polygons are similar polygons iff their corresponding angles are congruent and their corresponding side lengths are proportional. For them to be similar can also mean that they have the same shape but not necessarily the same size. k kkk - a scale factor is for describing how much a figure is enlarged or deduced. For dilation, transformation that changes size of a figure but not shape, with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a,b)  (ka, kb)

EX EX EX2 EX

*Indirect measures are any method that uses formulas, similar figures, or proportions to measure an object. To find indirect measures* with similar triangles, you have to follow some steps: 1.Convert the measurements to a single unit (if needed) 2.Find the similar triangles 3.Find the measurement you need... (use cross products property) This skill is important because if you were cutting down a tree near your house and you want to calculate if it is safe to do so, you may use this skill to find out the height of the tree.

4ft EXAMPLE 1 3ft 20ft h ¾ = 20/h 3h = 80 h =

5.3ft EXAMPLE 2 4ft 29ft h 4/5.3 = 29/h 4h = h =

6ft EXAMPLE 3 h 40ft 136ft h/6 = 40/ h = 240 h = 1.765

(4+3)/2(9+12)  7/21 =1/3 EX /  30/60 =1/2 EX3 Circumference 75 Circumference 37 75/37 Already in simplest form…

1.SINE – the sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. sinA = opposite leg/hypotenuse = a/c sinA = opposite leg/hypotenuse = a/c sinB = opposite leg/hypotenuse = b/c sinB = opposite leg/hypotenuse = b/c 2.COSINE – the cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. cosA = adjacent leg/hypotenuse = b/c cosA = adjacent leg/hypotenuse = b/c cosB = adjacent leg/hypotenuse = a/c cosB = adjacent leg/hypotenuse = a/c 3.TANGENT – the tangent of an angle is the ratio of the length of the leg opposite t he angle to the length of the leg adjacent to the angle. tanA = opposite leg/adjacent leg = a/b tanA = opposite leg/adjacent leg = a/b tanB = opposite leg/adjacent leg = b/a tanB = opposite leg/adjacent leg = b/a You can also reverse it into (sin-1, cos-1, or tan-1) S.O.H.C.A.H.T.O.A. SINE.OPPOSITE.HYPOTENUSE.COSINE.ADJACENT.HYPOTENUSE.TANGENT.OPPOSITE.ADJACENT A B C a c b

Since by the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with the same acute angle measure, and since a trigonometric ratio is a ratio of two sides of a right triangle, this can help us solve right triangles.

A B C EX1. SinB = 5/6 EX2. CosB = 4/5 EX3. TanB = 5/4 EX1. Sin-1B = 5/6 = EX2. Tan-1B = 5/4 = EX3. Cos-1B = 36.87

ELEVATION above ELEVATION – the angle of elevation is the angle formed by a horizontal line and a line of sight to somewhere above the line. This is important when you are in a watch tower watching a plane descend in order to give correct directions. DEPRESSION below DEPRESSION – the angle of depression is the angle formed by a horizontal line and a line of sight to somewhere below the line. This is important when you are a forest ranger and you need to stand high in an observation tower in order to see if a forest fire breaks out.

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