Similarity Review day 2

Transcription

1 Similarity Review day 2

2 DD, 2.5 ( ΔADB ) A D B Center (, ) Scale Factor = C' C 4 A' 2 A B B' 5

3 The line y = ½ x 2 is dilated by a scale factor of 2 and centered at the origin. Which equation represents the image of the line after the dilation? 1) y = x 1 2) y = ½ x 4 3) y = x 4 4) y = ¼ x 1. In the coordinate plane, line p has slope 8 and y-intercept (0,5). Line r is the result of dilating line p by a factor of 3 with center (0,3). What is the slope and y-intercept of line r? a) Line r has slope 5 and y-intercept (0,2). b) Line r has slope 8 and y-intercept (0,5). c) Line r has slope 8 and y-intercept (0,9). d) Line r has slope 11 and y-intercept (0,8).

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5 Similarity Similar Triangle Proportions and more Side Splitting Theorem A line drawn parallel to a side of a triangle intersecting the other sides divides the sides proportionally. Parallel Line Theorem A corollary of the side splitting theorem is that when three or more parallel lines are cut by two transversals then the segments intercepted on the transversal are proportional. Angle Bisector Theorem An angle bisector divides the side drawn to proportional to the other two sides.

6 Scale Factor scalefactor = side1 side2 Corresponding sides of similar triangles are proportional, the scale factor is the simplified ratio of the corresponding sides (or simplified ratio of the corresponding segments of the similar shapes). Perimeter Ratio perimeter1 side1 perimeter2 side2 = ratio perimeter = scale factor The ratio of the perimeter, circumference or any linear component of similar objects is equal to the scale factor. Area Ratio area1 side1 = area2 side2 2 ratio area = (scale factor) 2 The ratio of the areas of similar objects is the square of the scale factor. Volume Ratio volume1 side1 = volume2 side2 3 ratio volume = (scale factor) 3 The ratio of the volumes of similar objects is the cube of the scale factor.

7 Geometric Mean Is the mean proportional between two values such as: value1 = geometricmean geometricmean value2 Altitude to the hypotenuse proportions: The altitude is the mean proportional between the hypotenuse segments. hypotenuse segment1 = altitude altitude hypotenuse segment2 A leg is the mean proportional between the whole hypotenuse and the hypotenuse segment adjacent to the leg. whole hypotenuse = leg leg adj segment hypotenuse In proof: Ways to prove similar triangles: AA similarity Postulate (prove two pairs of corresponding angles congruent) SAS similarity Theorem (prove two pairs of corresponding sides proportional and the corresponding angles between the sides congruent) SSS similarity Theorem (prove three pairs of corresponding sides proportional) Other reasons used in similar triangles proofs: Similar triangles corresponding angles congruent Similar triangles corresponding sides proportional product of the means = product of the extremes

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10 8. Find w, x, y, and z. 9. Use the diagram below to answer each question. Segment AC is an angle bisector. a. If BC = 20, AB = 18, AD = 45 find CD. b. If AB = 12, AD = 15, and BD = 9, find CD. 10. The ratio of the surface areas of two similar triangular prisms iss 16:25. What is the ratio of their volumes? 11. A cylindrical pot with an 8 inch diameter holds 416 cubic inchess of water. How much water will fill a similar pot with a diameterr of 10 inches? 12. Quad ABCD ~ Quad HIJK a. If, AB = 8, JK = 12, and CD = 9, find HI. b. If m<b = 50, find m<i. c. If the perimeter of Quad HIJK is equal to 36, what is the perimeterr of Quad ABCD?

11 13. Find x A billboard at ground level has a support length of 26 feet that extends from the top of the billboard to the ground. A post that is 5 feet tall is attached to the support and is 4 feet from where the base of the support is attached to the ground. In the figure shown, the distance, in feet, from the base of the billboard to the base of the support is labeled x. Create an equation that can be used to determine x. Discuss any assumptions that should be made concerning the equation. Use your equation to find the value of x To the nearest tenth. Show your work and explain your answer.

12 16. Line segment AB with endpoints A(4, 16) and B(20, 4) lies in the coordinate plane. The segment will be dilated with a scale factor of ¾ with a center at the origin to create AB ' '. What will be the length of AB ' '? a) 15 b) 12 c) 5 d) The figure shows two semicircles with centers P and M. The semicircles are tangent to each other at point B, and DE is tangent to both semicircles at F and E. If PB = BC = 6, what is ED? a) 6 b) 48 c) 8 d) In the xy-coordinate plane, ABC has vertices A(-4, 6), B(2, 6), and C(2, 2). DEF is shown in the plane. What is the scale factor And the center of dilation that maps ABC to DEF? a) The scale factor is 2, and the center of dilation is point B. b) The scale factor is 2, and the center of dilation is the origin. c) The scale factor is ½, and the center of dilation is point B. d) The scale factor is ½, and the center of dilation is the origin.