If- = then ai = L-c b d

Transcription

1 I^otes: Ratios and Proportions the comparison of two numbers using C^i'i/t'S/'o^, written'^a to b", a:b, or ^ (b7^ 0) Proportion an equation that states two rdho are equal; ex: 7 = (b 0, d ^ 0) b a Cross Products In a proportion, the product of the extremes equals the product of the means: If- = then ai = L-c b d Ratios should be expressed in simplest form, and can be used to find missing dimensions In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined. Example 1: Find the ratio of the width to the length of the rectangle. Simplify the ratio. 12 cm 4 cm Proportions are made up of parts called means and extremes: extremes means ^t,; ^ means < ^ extremes Example 3: Example 4: 2y 8 4y 54 Vh-/6> - too J / o

2 Extended ratios compare more than two numbers. Example 5: Find the measures of the angles of the triangle, if the angle measures have a ratio of 1:7:10. /go' Example 6: The ratio of the side lengths of a triangle is 6:7:9. and its perimeter is 99 meters. What is the length of the longest side? scale &y><^e) Sometimes you will have to set the proportion, and then solve. Example 7: An apartment building is 90 ft tall and 55 ft wide. The scale model of the building is 11 in wide. How tall is the scale model? 52-

3 S.lNotes: Similar Polygons Congruence: Same shape and size (=) Have the same 5kaf>^ but not necessarily the same SiZe^ *xve use the symbol (~) to denote similarity Similar Figures: Corryesponding angles are ^ corresponding sides are Scale Factor/Similarity Ratio: the common ratio of the corresponding side lengths in similar figures. " ' " Example 1: AABC ~ ADEF AB DE to BC _ 4_ Z The scale factor/similarity ratio is: AC _ 5 / DF~ Cp ~ ^ Example 2: Identify the pairs of congruent angles and corresponding sides.. C = DF ^ 2> Z B = Z ^ EF y And by third angle theorem: Z A ^ Z P DE (o These triangles are similar because: What is the similarity ratio of: AABC and ADEF 3 '2 Write the similarity statement for the two triangles: A A^t/^ ~A

4 Example 3: ADEF ~ AMNP. Find x. N D T? F M 16 V Example 4: ABCDE-FGHJK. Find the scale factor (ratio): A 10 B Findx= 5v = "SC* '8 E 10 D Find the perimeter of ABCDE= JOJ^^ ^>o \i'2.'=-4lp K 15 Find the perimeter of FGHJK= t^*h*-n f^-/s - 6>^ What would the scale factor (ratio) of the perimeters be? Co'? ' What relationship do you see between the two scale factors (ratios)? Sc3/^ ^

5 Notes: Change in Dimension Similarity, Perimeter and Area Ratios Scale factor Ratio of Perimeter a:b a b Ratio of Area a' Example 1: ^ ~^ ^ Ratio of Corresponding Ratio of Perimeters Ratio of Areas Side lengths 4:5 7:^ 7:9 1^:^ 144:25 -it/se the figures below to answer questions # Find the scale factor of the sides. (EFGHMBC^) A 10 cm 15 cm 2. Find the perimeter of\^cd 4 cm 6 cm 3. Find the perimeter of EFCill 4. Find the scale factor of the perimeters (EFGH / ABCD) 5. How does the scale factor of the sides compare to the scale factor of the perimeter? 6. Find the a;;ea of ABCD ^ ' ^he^^ are. Me sar^e., 7. Find the area of EFGH 1 \ 8. Find the scale factor of the areas (EFGH/ABCD) 9. How does the scale factor 3f the sides com Dare to the sc'aie factor of the area? ^

6 Ex. 2 Find the scale factor of AEFG ~ AHJK: find the perimeter AHJK: find the area of AHJK: I %2 Ex 3 Tony and Edwin each built a rectangular garden. Tony's garden is twice as long and twice as wide as Edwin's garden. If the area of Edwin's garden is 600 square feet, what is the area of Tony's garden? C- I

7 Notes: Methods of proving Triangles Similar Rp'-ill what similarity means: 1] Corresponding angles are COnQroC*^ 2] The ratios of the measures of corresponding sides are. Angle- Angle Similarity (AA~) - If two angles of one triangle are =^. to two corresponding angles of another triangle, then the triangles are similar ^ ^^^^ Side- Side- Side Similarity (SSS~) - If the three sides of one triangle are ^'^g^cx- \\. / to the thr^ corresponding sides of another triangle, then the triangles are similar. In this example, the ratios of sides are:. a :x^ 6: 7.5 = 12: 15 = 4: 5. b:v = 8: 10 = 4:5 c: z = 4jj^ ' " ' These ratios are all equal, so the two triangles are similar. Side-Angle- Side Similarity (SAS~) - If two sides of one triangle are ^rp^p/' {' ong^ ( to two corresponding sides of another triangle and their included angles are ^ then the triangles are similar. In this example we can see that: one pair of sides is in the ratio of 21 : 14 = 3 : 2 another pair of sides is in the ratio of 15 : 10 = 3 : 2. there is a matching angle of 75 in between them *The 3 ways to prove similar triangles are:, and ^^^"^ 1^

8 Examples Decide if each pair of triangles is similar. If they are, write the correspondence in the first blank and the reason in the second blank. If they are NOT similar, write NS in the second blank. 1) A ABC-A 96(^ by -^^S ^ 7 2) A ABC-A S)i(^ by Ak^ B 3) A JKL ~ A J/A/ 4) ATXU-A ^/Xiy} by SJS Example 5: Explain why the triangles are similar, then find BE and CD + 6 Example 6: Find the value of x that makes AFGH ~ AJKL. (v-j) (y^i.^ i<2x ' 8o = *2x c /OS 2S

9 Notes Chapter 8.4 Properties of Similar Triangles and Proportional Relationships Triangle Proportionality Theorem Example If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides f^^ofori^^o^m. So BX _ BY ^ XA-YC ex 1: Find PN ex 2: Find US u, 14, Converse of the Triangle Proportionality Theorem Example If a line divides two sides of a trianqle,proportionally, then it is fhraflejtn the third side. ex - - Q L X4 vc ^ ex 3. Verify that DE// BC 7 IS 2^ 8 f 12 C Triangle Angle Bisector Theorem An 3nglA^''Sfic>4<'-,f p trianrjlp Hi\/idp«thp opposite side into tvyo segments whose the other two sides. (A zl Bisector Thm.) Example - ' VC 9 3 / ^ ^ ^ ^ X " " ^ /le _ 40 _ 5 A^^^r Ic AC 2A 3 ex.

10 Two Transversal Proportionality Corollary: If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. CE BD DF ex 4. Suppose that an artist decided to make a larger slcetch of the trees. In the figure, if AB = 4.5 in., BC = 2.6 in.. CD = 4.1 in., and KL = 4.9 in., find LM and MN to the nearest tenth of an inch, / i ^,/ j A/ore-.' V^fifi'S parejl&i 2.6 in M - J ^ " ' Che.' y t-^ "x^ Ex 5 : In Goodville Oak, Pine, Cedar and Maple Streets are parallel streets that intersect Main Street and Central Avenue, as shown in the diagram. Main St. A. How long to the nearest foot is Central Avenue from Pine Street to Cedar Street? Central Avenue B. How long is Central Avenue from Oak Street to Maple Street? ^ 580 r

11 Notes: Indirect measurement Ex. 1 A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? ^0 _ no " X /7o 14 ft 2 in. Ex 2 The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1in : 20ft. X scale (,0 3" / 10 "^^ Ex.3 In the figure, ADBA = AECA. What is the distance across the lake? 1^ Ex.4 Find the height of the tree. 4 7^ /2. E e 12 f1 CD 3 ft JL -

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