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Transcript
1 Chapter 11, Similarity Agenda: Review ratio and proportion Cover Chapter 11 this week Reminder: bring scientific calculator next time for Chapter 12 (trig) A ratio is an expression that compares two quantities by division, examples: a/b, a to b, a:b; we will use the fraction form. A proportion is a statement of equality between two ratios. Example A: A car travels 106 miles on 4 gallons of gas. How far can it go on a full tank of 12 gallons? Example B: Solve x 1 4 2 x 3 13 2 11.1 Similar Polygons Objective: learn what it means for two figures to be similar Use the definition of similarity to find missing measures in similar polygons Explore the dilations of figures on a coordinate plane Two polygons are similar if and only if The corresponding angles are congruent The corresponding sides are proportional Examples on the board A dilation is a non-rigid transformation of a figure which preserve the shape, but either stretch the sides or shrink the sides with same proportion. To dilate a figure in the coordinate plane, is to multiply the coordinates of all its vertices by the same number, called scale factor. Examples on the board Dilation Similarity Conjecture: If one polygon is dilated image of another polygon, then the polygons are: ___________________. Homework # 69, pp. 585-587 # 1-2, 4-18 even 3 11.2a Similar Triangles Objective: Learn shortcuts for determining whether two triangles are similar Remember the congruent shortcuts? Investigation: If two angles of one triangle are congruent to another triangle, must the two triangles be similar? AA Similarity Conjecture: If _________ angles of one triangle are congruent to _________ angles of another triangle, then the two triangles are __________________. (like the congruent shortcut: ASA, AAS) Investigation: If three sides of one triangle are proportional to the three sides of another triangle, must the two triangle be similar? SSS Similarity Conjecture: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are ______________. 4 11.2b Similar Triangles Investigation: Is SAS a similarity shortcut? What is SAS it means corresponding sides are proportional and the angle between the sides are congruent. SAS Similarity Conjecture: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the two triangles are similar. Examples on the board Is SSA a similarity shortcut? No, give a counter example. If you know two triangles are similar, then by definition of similarity, that the corresponding angles are congruent and the corresponding sides are proportional, i. e. ABCDEF, if and only A D, B E, C F, and AB BC CA DE DF FD More examples on the board 5 11.2c Similar Triangles If ABC is a right triangle, and CD is the altitude, then the segment CD divides the ABC into 3 similar right triangle, i.e. ABC ACD BCD, and CD2 = AD*BD C Example: A D B If ABC is a right triangle with C = 90°, B = 50°, what about ACD, BCD and B? If BD = 9, AD = 16, CD = ? BC = ? AC = ? Homework #70, pp. 591-592, #1-15 6 11.3 Indirect Measurement with Similar Triangles Objective: Learn how to use similar triangles to measure tall objects and large distances indirectly. Indirect measurement means to calculate the height of tall objects that you can’t reach using similar triangles. Example A: A person is 5’3” tall cast a shadow of 6’. A tree is tall and cast a shadow 18’. Find the height of the tree. Example B: A brick building casts a shadow 7’. At the same time, a 3’ child casts a shadow of 6” long. How tall is the building? C explain this on the board Example C: D A E B Homework # 71, pp 601-602, #1-7, 11-13 7 11.4 Corresponding Parts of Similar Triangles Objective: To investigate the relationship between two similar triangles Proportional Parts Conjecture: If two triangles are similar, then the lengths of the corresponding altitudes, medians, and angle bisectors are proportional. Reminder: An angle bisector is not necessary a median, and a median is not necessary an angle bisector. Angle Bisector/Opposite Side Conjecture: A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the two sides forming the angle. Example: Prove that the lengths of the corresponding medians of similar triangles are proportional to the lengths of the corresponding sides. HW#72, pp 605-607, 1-15 odd, 17 8 11.5 Proportions with Area Objective: to discover the relationship between the area of similar figures. Proportional Areas Conjecture: If corresponding side lengths of two similar polygons or the radii of two circles compare in the ratio m/n, then their areas compare in the ratio (m/n)2 or m2/n2 Example 1: If the area of the triangle ABC has proportion to the trapezoid BCDE is 1 to 24, find the ratio of AB/BD. Can this conjecture apply to the surface areas? Read the Investigation 2 on the Text. Example 2: If you need 3 oz of shredded cheese to cover a medium 12 in. diameter pizza, how much shredded cheese would you need to cover a 16 in. diameter pizza? HW#73, pp 610-612, 1-8, 10-16 even. 9 11.6 Proportions with Volume Objective: to discover the relationship between the volumes of similar figures. Example A: Are these right rectangular prisms similar? The smaller one has dimension 2, 3, and 7. The larger one has dimension 2, 6, 14. Example B: Are the right circular cones similar? The smaller one with radius 8cm, height 14, the larger one with radius 12, height 21. Example C: Are the two right cylinders similar? The smaller one with radius 3cm, height 7cm,and the larger one with base circumference 18cm, height 21cm. Proportional Volume Conjecture: If corresponding edge lengths (or radii, or heights) of two similar solids compare in the ratio m/n, then their areas compare in the ratio (m/n)3 or m3/n3 If the pentagonal pyramids are similar with h/H = 4/7, and the smaller pyramid has volume 320cm3, what is the volume of large pyramid? A “square can” is a right cylinder that has height equal to its diameter. One square can has height 5cm, and another has height 12 cm. About how many full cans of water from the smaller can are needed to fill the larger one. HW#74, pp 616-617, 1-11 odd. 10 11.7 Proportional Segments between Parallel Lines Objective: Discover the relationship in the lengths of segments formed when one or more lines parallel to one side of a triangle intersect the other two sides. Learn the applications of the relationships. Example A: Given ABC with the segment DE parallel to the side of AB, then ABC DEC (on the board) Example B: From example A, if CD = 48, DA = 36, CE = 60, then EB = ? Parallel/Proportionality Conjecture: If a line parallel to one side of a triangle pass through the other two sides, then it divides the other two sides proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to the third side. Examples on the board. Extended parallel/Proportionality Conjecture: IF two or more lines pass through two sides of a triangle parallel to the third, then they divide the two sides proportionally. Examples on the board HW#75, pp 627-629, 1-21 odd. 11