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Transcript
Chapter 7 Section 7.1 β Ratios and Proportions Objectives: To write ratios and solve proportions οͺ Ratio -> comparison of two quantities π οͺ This can be written as a : b or when b β 0 (for our purposes we π will assume all terms and expressions are nonzero) οͺ Proportion -> a statement that two ratios are equal οͺ Can be written is these forms: π π π π = and a : b = c : d οͺ Extended Proportion -> when three or more ratios are equal οͺ Properties of Proportions οͺ π π π = π is equivalent to: 1. ad = bc 2. 3. 4. π π = π π π π = π π π+π π+π = π π means a:b=c:d extremes Cross Product Property -> also know as βthe product of the extremes is equal to the product of the meansβ οͺ Scale drawing -> a given scale compares each length in the drawing to the actual length. The lengths used may be different units, possibly written as 1 in. to 100 mi, 1 in. = 12 ft, or 1 mm : 1 m. οͺ Examples: οͺ Solve the following proportions π₯ 5 12 = 7 5 20 = π§ 3 18 6 = π+6 π οͺ Write two proportions that are equivalent to: π₯ π¦ = 5 6 π 4 = π 11 οͺHomework # 1 οͺDue Wednesday (January 09) οͺPage 368 β 369 οͺ# 1 β 19 odd οͺ# 35 β 42 all Section 7.2 β Similar Polygons οͺ Objectives: To identify similar polygons To apply similar polygons οͺ Similar (~) -> two figures that have the same shape but not necessarily the same size οͺ 1. Corresponding angles must be congruent οͺ 2. Corresponding sides are proportional οͺ Similarity Ratio -> the ratio of the lengths of corresponding sides οͺ ABCD ~ EFGH. Complete each statement. B C F G 127° E 53° A D H π΄π΅ πΈπΉ m<E=? m<B=? = π΄π· ? οͺ Determine whether the triangles are similar. If they are, write a similarity statement and give the similarity ratio. B 15 A 12 18 E C 20 16 D 24 F οͺ LMNO ~ QRST O 2 Find the value of x. N T 3.2 L 5 x S M Q 6 R οͺ Golden Rectangle -> a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle. οͺ Golden Ratio -> the ratio in any golden rectangle about 1.618 : 1 οͺHomework #2 οͺDue Friday (January 11) οͺPage 375 β 376 οͺ# 1 β 16 all οͺ# 21 β 29 odd Section 7.3 β Proving Triangles Similar οͺ Objectives: To use AA, SAS, and SSS similarity statements To apply AA, SAS, and SSS similarity statements οͺ Postulate 7.1 οͺ Angle-Angle Similarity (AA ~) Postulate οͺ If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. P T R S L M οͺ Theorem 7.1 -> Side-Angle-Side Similarity (SAS ~) Theorem οͺ If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar. οͺ Theorem 7.2 -> Side-Side-Side Similarity (SSS ~) Theorem οͺ If all three corresponding sides of two triangles are proportional, then the triangles are similar. οͺ Ex: Explain why the triangles must be similar. Write a similarity statement. Z P 6 R 8 3 Q Y 4 X οͺ Ex: Explain why the triangles must be similar. Write a similarity statement. 9 A 6 B G 6 8 8 C E 12 F οͺHomework # 3 οͺDue Monday (January 14) οͺPage 385 β 387 οͺ# 1 β 19 odd οͺ# 24 β 27 all Section 7.4 β Similarity in Right Triangles οͺ Objectives: To find and use relationships in similar right triangles οͺ Theorem 7.3 οͺ The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. C A Triangle ABC ~ Triangle ACD ~ Triangle CBD B D οͺ Geometric Mean -> proportions in which the means are equal. For any two positive numbers a and b, the geometric mean of π π₯ a and b is the positive number x such that = π₯ π **We will cover two corollaries that involve a geometric mean** οͺ Corollary 1 to Theorem 7.3 οͺ The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. C π΄π· πΆπ· A D B πΆπ· = π·π΅ οͺ Corollary 2 to Theorem 7.3 οͺ The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse. C A D B π΄π· π΄πΆ π΄πΆ = π΄π΅ π·π΅ πΆπ΅ πΆπ΅ = π΄π΅ οͺHomework #4 οͺDue Tuesday (January 15) οͺPage 394 οͺ# 1 β 20 all οͺ# 26 β 33 all Section 7.5 β Proportions in Triangles οͺ Objectives: To use the Side-Splitter Theorem To use the Triangle-Angle-Bisector Theorem οͺ Theorem 7.4 -> Side-Splitter Theorem οͺ If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Q R X S ππ π π Y ππ = ππ οͺ Corollary to Theorem 7.4 οͺ If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. π π a b c d π = π οͺ Theorem 7.5 -> Triangle-Angle-Bisector Theorem οͺ If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. πΆπ· π·π΅ A B C D πΆπ΄ = π΅π΄ οͺHomework # 5 οͺDue Wednesday (Jan 16) οͺPage 400 β 401 οͺ# 1 β 24 all οͺQuiz Friday οͺTest Thursday/Friday next week