Math Chapter 6 Notes

6.1 Ratios, Proportions, and the Geometric Mean VOCABULARY Ratio of a to b Proportion Means and Extremes Geometric Mean EX1: Simplify Ratios Simplify the ratio. (See Table of Measures, p. 921) a. 76 cm: 8 cm b. 4 meters to 18 meters c. 4 ft 24 in d. AB: AC EX2: Use a Ratio to Find a Dimension You are painting a barn door. You know that the perimeter of the door is 64 feet and that the ratio of the length to the height is 3:5. Find the length and width of the door, and then find the area of the door. EX3: Use Extended Ratios a. The measures of the angles in BCD are in the extended ratio of 1:4:5. Find the measures of the angles. b. Find the measures of the angles. A PROPERTY OF PROPORTIONS Cross Products Property In a proportion, the product of the extremes equals the product of the means. If a c  b d EX4: Solving Proportions 3 x a.  5 16 b. 3 2  x 1 x c. RS:RT = 13:25 GEOMETRIC MEAN: When you set up a proportion to find the geometric mean, put an x in both of the spots that are considered the “means” of the proportion. You will always use a _______ when solving for x. Find the geometric mean of the two numbers. a. 6 and 24 b. 14 and 16 The three coordinate points are collinear. Use slopes to write a proportion to find the value of a. a. (4, 5), (1, 2), (a, 0) b. (–4,1), (–1,2), (5, a) 6.2 Use Proportions to Solve Geometry Problems VOCABULARY A scale drawing is a drawing that is the same shape as the object it represents. Scale Drawing Scale The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object. EX1: Find the Scale of a Drawing Keys The length of the key in the scale drawing is 7 centimeters. The length of the actual key is 4 centimeters. What is the scale of the drawing? To find the scale, write the ratio of a length in the drawing to the actual length, then rewrite the ratio so that the smaller number (denominator) is 1. The scale of the drawing is ________________. EX 2: Use a Scale to set up a proportion Scale Model You buy a 3-D scale model of the Sunsphere in Knoxville, TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the diameter of the dome on your scale model is about 5.6 inches. What is the diameter of the actual dome? Scale Map A map is scaled so that 1 cm represents 20 km. If two towns are 8.2 centimeters apart on the map, what is the actual distance between the towns? ADDITIONAL PROPERTIES OF PROPORTIONS (Just copy the format of the first ratio . . .) Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If a c  b d , then b  a . If you interchange the means of a proportion, then you form another true proportion. a c a   If , then . b d c In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another true proportion. If , then a b  c a  b d b .  EX3: Use Properties of Proportions In the diagram, AC DF  BC . Find the value of x. EF EX4: Use Proportions with Geometric Figures If JL LH JK  KK , find JH and JL. KG 6.3 Use Similar Polygons VOCABULARY Similar Polygons Scale Factor of Similar Polygons EX1: Use Similarity Statements In the diagram, ∆ ABC  ∆ DEF. a. List all pairs of congruent angles. c. b. Check that the ratios of corresponding side lengths are equal. Write the ratios of the corresponding side lengths in a statement of proportionality. In a statement of proportionality, any pair of ratios forms a true proportion. EX2: Find the Scale Factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKLM. EX3: Use Similar Polygons a. In the diagram, ∆BCD  ∆RST. Find the scale factor of BCD to RST and the value of x. b. In the diagram, LMNP  FGHJ. Find the scale factor of FGHJ to LMNP and the value of x. THEOREM: PERIMETERS OF SIMILAR POLYGONS If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their SCALE FACTOR. If KLMN  PQRS, then EX 4: Find Perimeters of Similar Figures Basketball A larger cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 25 feet long. The old court was similar in shape, but only 16 feet wide. a. Find the scale factor of the new court to the old court. b. Find the perimeters of the new court and the old court. Now try this one on your own! In the diagrams, ∆PQR  ∆WXY. Find the perimeter of ∆WXY. 6.6 Use Proportionality Theorems THEOREM: TRIANGLE PROPORTIONALITY THEOREM If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides ________________________________. If QS || TU , then _________= _________ THEOREM:CONVERSE OF THE TRIANGLE PROPORTIONALITY THEOREM If a line divides two sides of a triangle proportionally, then it is parallel to the ________ . If RT RU , then ______ || ______  TQ US EX 1: Find the Length of a Segment In the diagram, QS || UT , RQ = 10, RS = 12, and ST = 6. What is the length of ? QU More examples… a. Find the length of KL b. Determine whether QT || RS. EXPLAIN THEOREM: 3 OR MORE LINES CUT BY TRANSVERSALS If three parallel lines intersect two transversals, then they divide the transversals ___________________ EX2: Using the Theorems about Proportionality with parallel lines a. Farming A farmer’s land is divided by a newly constructed interstate. The distances shown are in meters. Find the distance CA between the north border and the south border of the farmer’s land. The distance between the north border and the south border (FD) is ___________ meters. b. Find the length of AB . THEOREM: AN ANGLE OF A TRIANGLE IS BISECTED If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are ___________________ to the lengths of the other two sides. AD DB = _____ EX3: Using the Theorems about Proportionality with an angle of a triangle bisected a. b. Find the value of x. 6.4 & 6.5 Prove Triangles Similar by AA~, SSS~, and SAS~ POSTULATE: ANGLE-ANGLE (AA~) SIMILARITY POSTULATE If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. ∆JKL ~ ∆XYZ EX 1: Use the AA Similarity Postulate Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. a. b. c. EX 2: Show that Triangles are Similar Show that the two triangles are similar. a. ∆RTV ~ ∆RQS b. ∆LMN ~ ∆NOP c. ∆BCD ~ ∆EFD EX 3: Using Similar Triangles A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair? THEOREM: SIDE-SIDE-SIDE (SSS~) SIMILARITY THEOREM If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If AB BC CA   , then ABC ~ RST. RS ST TR THEOREM: SIDE-ANGLE-SIDE (SAS~) SIMILARITY THEOREM If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If X  M , and EX 4: Use the SSS Similarity Theorem Is either DEF or GHJ similar to ABC? ZX XY  , then XYZ  MNP. PM MN When using the SSS Similarity Theorem, compare the shortest sides, the longest sides, and then the remaining sides. CHECKPOINT: Are the triangles similar? If so, state the theorem/postulate and the similarity stmt. 6.7 Perform Similarity Transformations VOCABULARY Dilation (Center) Scale Factor (k) Reduction 0k1 Enlargement k1 COORDINATE NOTATION FOR A DILATION You can describe a dilation with respect to the origin with the notation (x, y)  (kx, ky), where k is the scale factor. EX1: Draw a Dilation with a Scale Factor Greater Than 1 Draw a dilation of quadrilateral ABCD with vertices A(2, 0), B(6, 4), C(8, 2), and D(6, 4). 1 Use a scale is factor of . Draw ABCD, then draw the dilation A’B’C’D’. 2 (x, y)  ( ___x, ___y) All of the dilations in this lesson are in the coordinate plane and each center of dilation is the origin. A(2, 0)  A’ _______ B(6, 4)  B’ _______ C(8, 2)  C’ _______ D(6, 4)  D’________ EX2: A Figure is Similar to its Dilation A triangle has the vertices A(2, 1), B(4, 1), and C(4, 2). The image of ABC after a dilation with a scale factor of 2 is DEF. Sketch ABC and DEF. EX3: Find a Scale Factor Magnets You are making your own photo magnets. Your photo is 8 inches by 10 inches. The image on the magnet is 2.8 inches by 3.5 inches. What is the scale factor of the reduction? EX4: Find Missing Coordinates You want to create a quadrilateral JKLM that is similar to quadrilateral PQRS. What are the coordinates of M? Solution Determine if JKLM is a dilation of PQRS by checking whether the same scale factor can be used to obtain J, K, and L from P, Q, and R. (x, y)  (kx, ky) k  ___ P(______)  J(______) Q(______)  K(______) k  ___ R(______)  L(______) k  ___