Chapter 7 & 8 Kirsten Erichsen Journal Geometry

RATIOS AND PROPORTIONS

What is a ratio?  A ratio is used to compare 2 numbers by a division.  Each ratio can involve more than two numbers.  It can be written using a and b as the following:  a to b  a:b  a/b

Examples of a Ratio.  Find the ratio for the following shapes Ratio = 4:8:4: Ratio = 5 :3: Ratio = 2 :2:2:2

What is a Proportion?  A proportion is an equation that states that 2 ratios are equal.  Extremes:  A/B = C/D (A and D are the extremes of the proportion)  Means:  A/B = C/D (B and C are the means of the proportion)

How to solve a Proportion.  First you have to cross multiply the numbers and the variable. After you are done multiplying, you have to leave the variable alone to find its value that’s why you have to divide.  If you have 2 variables in a proportion, you have to square root both sides.  To check your proportions, you multiply the extremes together and the means together (opposite sides) and see if they are the same. If you have it different you might have done something wrong, or it is not a proportion.

Examples of a Proportion. Example 1: 3 = X 6 = 8 3 × 8 = 24 6X = 24 Example 2: 7 = 4 + Y 10 – Y = 7 7 × 7 = – 3 = = 7 Example 3: 8 = X X = 8 8 × 8 = 64 √X 2 = √64 X = 4Y = 3X = 8

Proportions and Ratios.  A proportion is related to a ratio because a proportion is an equation where 2 ratios are equal  EXAMPLES:  A teacher counted boys to girls and ended with a ratio of 5:4. There were fifteen boys, but how many girls where there?  A Vet has patients, he has a ratio of dogs to cats per day, 6:4. His assistant counted 24 dogs, how many cats where there?  An Architect has a ratio of blocks to plastic tubes, 15:4. He counted 60 blocks, how many plastic tubes where there? Answers: 1.) 12 girls; 2.) 16 cats; 3.) 16 plastic tubes

RATIOS IN SIMILAR POLYGONS

Similarity in Polygons.  This is when polygons have the same shape, but not necessarily the same size.  Two polygons are similar if and only if, their corresponding angles are congruent and their corresponding side lengths are proportional.

Example 1.  Find the missing side length Since both sides in the smaller triangle have half of the measurements of the bigger triangle, we just multiply 8 times 2, so x will equal x

Example 2.  Find the missing side length. 10 Since both sides in the smaller rectangle have half of the measurements of the bigger rectangle, we just multiply 20 times 2, so x will equal X 20

Example 3.  Find the missing side length x 6 8 Since both sides in the smaller trapezoid have half of the measurements of the bigger trapezoid, we just multiply 3 times 2, so x will equal 6.

What is a Scale Factor?  A scale factor describes how much a figure has been enlarged or reduced.  Dilation: transformation that changes the size of the figure but not its shape.  If the scale factor of a dilation is greater than 1 it is an enlargement.  If the the scale factor is less than one, it is a reduction.

Example 1.  Find the ratio of AB to KL. A 5 C 5 AB to KL = 5 to B K L M

Example 2.  Find the ratio of NP to QS. N 6 P 6 NP to QS= 1 0 to 5 10 O Q R S

Example 3.  Find the ratio of WY to TV. T 8 V 8 WY to TV= 9 to U W X Y

INDIRECT MEASUREMENTS

How to use similar triangles to perform indirect measurements.  Indirect Measurement: it is any method that uses formulas, similar figures, or proportions to measure an object.  When using similar triangles to perform indirect measures you have to use proportions to find your answer. Use the actual height and the measure of shadows.  This is an important skill because that way in real life you can find the exact measurement of something very tall, like a building.

Example 1.  A tourist travels to the Egyptian pyramids, he finds out that their height is unknown. He measures 1.80 meters. The shadow of the pyramid is 20 meters, and his shadow measures 1.75 meters. What is the height of the pyramid? 20 meters 1.75 meters 1.80 meters 20 = 1.80 X = X = 35 X = meters X

Example 2.  An architect wants to find out the measure of a building to build one exactly the same. His shadow measures 1.90 meters and his height is 2 meters. The building’s shadow measures 40 meters, can you tell me what the height of the building is? 40 meters 1.90 meters 2 meters 40 = 2.00 X = X = 76 X = 38 meters X

Example 3.  A pilot wants to find the height of his airplane. He measures 1.90 meters, his shadow measures 1.80 meters. The shadow of the airplane measures 25 meters. What is the airplanes height? 25 meters 1.80 meters 1.90 meters 25 = 1.90 X = X = 45 X = 23.7 meters X

SCALE FACTOR, PERIMETER & AREA

How to use the scale factor to find the perimeter and area.  PERIMETER: To find the perimeter using scale factor, you first need to find the perimeter of each triangle. Then you create a fraction with each perimeter, but remember, it is the smaller shape over the bigger shape. Then you simplify the fraction. The ratio of the perimeter is the same as the ratio of their sides.  AREA: To find the area using scale factor, you first need to find the area of each shape. You make a fraction out of those areas (small shape over the bigger shape). After you have made the fraction, you simplify it all the way and then you square it.

Example 1.  Find the perimeter and area for each shape and then use the scale factor. PERIMETER: 1 6 = 2 24 = 3 AREA: 1 6 = =

Example 2.  Find the perimeter and area for each shape and then use the scale factor. PERIMETER: 8 17 AREA:

Example 3.  Find the perimeter and area for each shape and then use the scale factor. PERIMETER: 1 2 = 1 24 = 2 AREA: = =

TRIGONOMETRIC RATIOS

What are the 3 ratios?  Ratio One (SIN): 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  Ratio Two (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  Ratio Three (Tangent): the tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle.  tanA: opposite leg adjacent leg

Example 1.  Find the 3 trigonometric ratios for the shape. Sin: 5/13 Cos: 12/13 Tan: 5/

Example 2.  Find the 3 trigonometric ratios for the shape. Sin: 3/10 Cos: 8/10 Tan: 3/

Example 3.  Find the 3 trigonometric ratios for the shape. Sin: 9/15 Cos: 10/15 Tan: 9/

How are they used to solve for right triangles?  What does it mean to solve a right triangle? Well, to solve for a right triangle, states that you have to find the measurements of each side and angle.  How to find the angles?  First, you need to know what type of ratio it is.  Then you plug in the the inverse of the ratio into your calculator (Ex. Sin-1(12/13) = 66.92)  That is how you get your angle measures.  How to find the sides?  To find the side, you need to find the type of ratio.  Then, you make a proportions with the missing value and the other side.  You plug in the side measure and then the ratio with the angle measure between those sides. (Ex. Tan40 = X; 100(tan40) = 83.9

Example 1.  Find the missing measurement for letter x. Sin42 = x/12 12(sin42) = x X = 8.02 X 12 42

Example 2.  Find the missing measurement for letter X. 8 X 42 Sin42 = 8/x 8 = xSin42/sin42 X = 11.96

Example 3.  Find the missing measurement for letter X X Sin(12/13) Sin-1(12/13) X = 67.38

ANGLE OF ELEVATION AND DEPRESSION

Angles of Elevation and Depression.  Angle of Elevation: it is the angle formed by a horizontal line and a line of sight to a point above the line. Watch from down to up.  Angle of Depression: the angle formed by a horizontal line and a line of sight into a point below the line. Watch from up to down. Angle of Depression Angle of Elevation

Example 1.  Find the missing measurement for X X Tan32 = 2/X X = 2/tan32 X = 3.2 meters

Example 2.  Find the missing measurement for X X Tan25 = 5/X X = 5/tan25 X = 4.3 meters

Example 3.  Find the missing measurement for X X Tan33 = 9/X X = 9/tan33 X = meters