What is a ratio? What is a proportion? Give an example of two ratios that reduce to the same value
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1 Geometry A Chapter Ratio and Proportion What is a ratio? What is a proportion? Give an example of two ratios that reduce to the same value How do you solve a proportion? ex: 3x + 2 = 5x
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7 In a proportion, the first and fourth terms are called the extremes and the second and third terms are called the means Theorem 59: Means-Extremes Products Theorem: the product of the means is equal to the product of the extremes. THEOREM 60: If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means of a proportion. (MEANS-EXTREMES RATIO THEOREM Find the ratio of x: y if 5x-6by=2cx+ya
8 Definition: If the means in a proportion are equal, then either mean is called a geometric mean or mean proportional between the extremes. 8/12=12/18 1/5=5/25 Can you find an example of a geometric means proportion? Find the geometric and arithmetic means between 6 and 36
9 8.2 Similar Polygons def: Similar polygons are polygons in which 1) the ratios of the corresponding sides are equal and 2) the measures of the corresponding angles are congruent. Can you create a figure that is three times as large as the given figure?
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11 Two figures are similar if one is a reduction or a dilation of the other D (0, 6) C A (0, 0) B(8,0) E ΔAED is a dilation of ΔABC in the ratio of 2:3. Find the lengths of sides of ΔAED (recall that in a right Δ, a 2 +b 2 =c 2
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13 2 5 Let these triangles be similar with a ratio of 3 to 5 x z 4 y Find the length of the missing sides x, y, z AND find the perimeters of both triangles and their ratio
14 If two polygons are similar, then the ratio of their sides is equal to the ratio of their perimeters y 12 x what is the ratio of perimeters? Can you find the perimeter of the second triangle without first determining the lengths of the sides?
15 10/18 = 1010/k k=1818, h = 1820
16 Solve for x and y, given that the triangles are similar
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23 similar triangles D B B E A C
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25 Assume that you are 5 ft. 4 in. tall and that you are standing 10 ft from the mirror. Assume that the mirror is placed 18 ft from the foot of the tree.. Set up a proportion to determine the height of the tree.
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27 G A 4 5 R ABC ~ 12 GDR D C 3 B Find all sides, find perimeters, find areas, find ratio of sides, ratio of perimeters and ratio of areas.
28 Congruences and proportions in similar triangles 8.4 If two triangles are congruent then their corresponding parts are congruent- CPCTC IF two triangles are similar then their corresponding parts are similar? NO IF two triangles are similar then (1) the corresponding sides are proportional (ie. THE RATIOS OF THE MEASURES OF CORRESPONDING SIDES ARE EQUAL) CSSTP (2) the corresponding angles are congruent CASTC Important ideas: product of means = product of extremes geometric mean: a/x = x/b, x is the gm AA~, SAS~, SSS~ ratio of sides = ratio of perimeters, BUT ratio of areas is the square of the ratio of sides,
29 B E A C D F GIVEN: <A = <D, <B = <E PROVE: AB* EF = BC * DE
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31 9 G 8 L M R J 15 K Q P GJKL ~ MQPR find QP. PR, MR
32 8.5 Three Theorems Involving Proportions Th 65: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally. called the SIDE-SPLITTER THM G: BE // CD P: AB/BC = AE/ED
33 TH 66: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally. A V G: AV // BW // CY B W P: AB/BC = VW/WY C Y Can you see how to prove this using side-splitter?
34 Th 67: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. (Angle Bisector Theorem) A Given: AD bisects <CAD C 1 2 D B Prove: AC/CD =AB/DB <1 = <2 1 2
35 3 V 5 4 G: <3 = <5 P: RV = RS VT ST R S T
36 x Find the perimeter of the largest triangle 5 9 x + 3 y
37 G: ABDF is a parallelogram P: CBD ~ DFE A B C F D E
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