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Transcript
McDougal Geometry chapter 6 notes 6.1 Ratios, proportions, and the geometric mean We will use first parts of TK#50 in 6.1 and the rest in 6.2. TK#50: Properties of proportions The geometric mean of 2 positive numbers a and b is the positive number x such that a x . x b 1. So x2=ab and x ab. Cross product property. In a proportion, the product of the extremes equals the product of the means. If 2. where b≠0 and d≠0, then ad=bc. Reciprocal property. If 2 ratios are equal, then their reciprocals are also equal. If 3. a c b d a c b d , then b d . a c If we swap the means of a proportion, then we form another true proportion. If a c b d , then a b . c d In a proportion, if we add the value of each ratio’s denominator to its numerator, then we form another true proportion. If a c b d , then ab cd . b d GP 2-4,6,7,11 Hwk: p360#7-65 odds (skip 55). 1 6.2 Use proportions to solve geometry problems, pg. 367 #3-17 odds, 18, 23-29 Example 1 and GP1. GP2. Example 3 and GP3. GP4. GP5. (If time) Prove: If a c ab cd , then . b d b d 2 6.3 Use similar polygons 2 polygons are similar (~) if corresponding angles are congruent and corresponding sides are proportional. If ABCD~EFGH, then corresponding angles are congruent: A E , B F , C G, D H and corresponding sides are AB BC CD DA . In writing the similarity statements, order is proportional: EF FG GH HE crucial! B F A D E G H C TK#51: Similar polygons Any corresponding 1-dimensional part (length, perimeter, radius, etc) of similar polygons is equal to the scale factor. Guided practice 1,4-6. 6.3 one day pg. 376#5-27 odds, 31, 36 or Section 6.3 day 1 pg. 376 #1-11, 40-45 Section 6.3 day 2 pg. 376 #12-26, 31-32 __________ Quiz Review pg. 380 #1-7 __________Quiz 6.1-6.3. Hwk: Activity on the top of pg 381: Angles and similar triangles (need protractor and ruler). Steps 1-4. Draw conclusions 1,2. 3 6.4 Prove triangles similar by AA, pg 384#3-25 odds, 32-36E Turn in Activity 6.4. TK#52: AA similarity postulate: If 2 angles in one triangle are congruent to 2 angles in another triangle, then the 2 triangles are similar. Since J X , K Y , then ∆JKL~∆XYZ by AA∆~Post. K Y L Z X J Guided practice 2, 4. 4 6.5 Prove triangles similar by SSS and SAS TK#53: SSS Similarity Thm: If all 3 corresponding sides of two triangles are proportional, then the triangles are similar. If AB BC CD , then ∆ABC~∆RST RS ST TR by SSS∆~ Thm. A R C T S B TK#54: SAS Similarity Thm: If 2 sides of one triangle are proportional to the corresponding 2 sides of another triangle and the included angles are congruent, then the two triangles are similar. If JK KL and K Y , then ∆JKL~∆XYZ by XY YZ SAS∆~ Thm. K Y L Z X J Find x to make ∆JKL~∆MYZ. Example K 3(x-2) Y 12 L 21 Z x+6 20 30 M J Guided practice 1-3. 5 If time: prove guided practice 4 using 2 different methods. 6.5 one day: pg 391 #3-23 odds, 29-35 odds or 6.5 day 1 pg 391 #3-23 odds, 39-44 (bring compass and straightedge tomorrow) 6.5 day 2 pg 393 #18-24E, 28-36E (bring compass and straightedge tomorrow) 6 6.6 Use proportionality thm, pg 400 #3, 9-18 TK#55: Angle Bisector Proportionality Thm: If a ray bisects an angle of a triangle, then it divides the opposite side into the same proportion as the 2 sides that form the angle. AD AC DB BC A D C B Example 4. Dilation Activity 6.7 on pg 408 Steps 1-4. Draw conclusions 1,2. (Need graph paper, compass, and ruler.) Hwk: Ch 6 review pg 418 #1-18. Print out Ch 7 notes. 7 6.7 Perform similarity transformations (Need graph paper). Pg 412 #3-21 odds (skip 17) Turn in: Activity 6.7, ch 6 review, and printed ch 7 notes. TK#56 Similarity transformation: With dilation: If 0<k<1, reduction. If k>1, enlargement. Guided practice 2, 3. _______ Ch 6 Group Quiz. Collect TK#1-56. _______ Correct ch 6 gp quiz. _______ Ch 6 individual Test 8