Geometry, 8.1: Ratio and Proportion
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1 Geometry, 8.1: Ratio and Proportion Ratio examples: Model car: Recipe: Mix: 1 gallon water The juice from 2 lemons 2 cups sugar This makes 1 gallon of lemonade. What would you mix if you needed to make 3 gallons of lemonade? Model: length = 1 foot Real car: length = 14 feet If side window on model is 4 inches long, how long is window on real car? Ratio: shows a relation of two numbers. Can be written different ways: Examples of ratios: slope (change in y over change in x) map scale (1 inch on map : 100 miles actual distance) recipes (2 lemons per gallon water, 2 cups sugar per gallon of water) speed (miles per hour) shapes of geometric figures (one side twice as long as another side)
2 Proportion: An equation stating that two ratios that are equal to each other. If you know all but one number of a proportion, you can solve to find the missing value: Solve by cross-multiplying (book calls this means-extremes product theorem) If you have an equation multiplied out, you can turn it back into a proportion... Find the ratio of x to y if 2x=3y: Arithmetic and Geometric mean: If you have a proportion that looks like this: 1 = with number on bottom of one ratio = number on top of other ratio, then: the proportion is called a mean proportion the common number is called the geometric mean Mean, in general, is a number that describes a group of other numbers. There are 2 kinds of 'means': Arithmetic mean To compute: Add numbers, and divide by 2. Example: Find arithmetic mean of 8 and 18: Geometric mean To compute: Make a mean proportion with x as geometric mean, then cross-multiply to solve for x. Example: Find geometric mean of 8 and 18:
3 Practice: #1. Find x: 3 x = #2. Find b: = 30 b #3. Find ratio of x to y if 4x=5y #4. Find x: 4 6 = x #5. A 20ft steel pole is cut into two parts in the ratio of 3 to 2. How much longer is the longer part than the shorter part?
4 Geometry, 8.2: Similarity Congruent figures Similar figures Figures are congruent when: Figures are similar when: 1. Corresponding sides are congruent. 1. Corresponding sides are. 2. Corresponding angles are congruent. 2. Corresponding angles are. Two ways a figure similar to another figure can be produced: Dilation = Reduction = Symbol for 'similar' is ~ example: ABC DEF Two figures can be proved similar if: Practice: Which pairs of polygons can be proved to be similar?
5 Practice: Given: NPR STV with lengths, angles in diagram Find: m T, m S, and VT Practice: The roof of a house has a slope of 5. What is the width of the house if the height of 12 the roof is 8 ft? ? Extra Information How to find the scale factor of two similar polygons: EFGH is a dilation of ABCD. What is the scale factor? What is the ratio of the perimeters of ABCD and EFGH?
6 Geometry, 8.3: Proving Triangles Similar As with proving triangles congruent, we have shortcuts to prove triangles are similar: Triangles Similar if: AAA All corresponding angles are congruent. AA Two corresponding angles are congruent. SSS~ The ratios of side lengths are equal for all pairs of corresponding sides. SAS~ The ratios of side lengths are equal for 2 pairs corresponding sides and the angle between is congruent. Practice: Are the pairs of triangles below similar? If so, by which shortcut?
7 Practice: Draw a triangle GJK. Then indicate a point H on GJ and a point M on GK such that HM JK. Prove that GHM GJK.
8 Geometry, 8.4: Congruence and Proportion in Similar Triangles Once we proved triangles congruent, we could prove angles or sides congruent by Once we prove triangles are similar we can prove: If a problem asks you to prove that products of sides are equal use: Example: Given: ABC DEF Prove: A D Example: Given: ABC DEF Prove: AB = AC DE DF Example: Given: ABC DEF Prove: AB DF = AC DE
9 Practice:
10 Geometry, 8.5: Three Theorems Involving Proportions Side-splitter theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. Example: Given: BE CD, lengths as shown Find: ED and CD Example: Solve for x and y (parallel lines / transversals theorem): If 3 or more parallel lines are intersected by 2 transversals, the parallel lines divide the transversals proportionally. Example: Find x
11 (Angle Bisector theorem): If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. Practice:
What is a ratio? What is a proportion? Give an example of two ratios that reduce to the same value
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