Lesson 3.1. Midpoint and Dilations (5.1.1, 5.2.1, and 5.2.2) Pages 8-38 in Unit 5 Workbook
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1 Lesson 3.1 Midpoint and Dilations (5.1.1, 5.2.1, and 5.2.2) Pages 8-38 in Unit 5 Workbook
2 midpoint What is a midpoint? (page 8) A point that is exactly ½ the distance between 2 endpoints. Look at the graph below: The endpoints are (-2, -6) and (4, 8). The MIDPOINT is at (1, 1)
3 midpoint To find a point that is a midpoint or half ( 1 ) the distance between two 2 endpoints: 1. Calculate the average between the x-values 2. Calculate the average between the y-values Midpoint: ( x 1+x 2 2, y 1+y 2 ) 2
4 midpoint Find the midpoint of the line segment with endpoints ( 2,1) and (4,10) (page 9)
5 endpoint To find an endpoint given another endpoint and the midpoint of the segment. 1. Add 2 times the difference of the midpoint x-value to the x-value of the given endpoint. a. Or add the difference of the midpoint x-value to the x-value of the midpoint. 2. Add 2 times the difference of the midpoint y-value to the y-value of the given endpoint. a. Or add the difference of the midpoint y-value to the y-value of the midpoint.
6 midpoint A line segment has one endpoint at (12,0) and a midpoint of (10, 2). Locate the second endpoint. (page 10 example 5)
7 midpoint A line segment has one endpoint at (3,5) and a midpoint of ( 3 2, 25 2 ). Locate the second endpoint. (not in WB)
8 dilation (page 18) A DILATION is a transformation in which a figure is either enlarged or reduced by a scale factor (k) Dilation Stretch
9 CENTER OF DILATION is the point about which all points are stretched or compressed. SCALE FACTOR of a figure is a multiple of the lengths of the sides from one figure to the other. A scale factor of 2 (k = 2) means that each side length will enlarge to be twice the length between the two figures. A scale factor of 1 3 (k = 1 3 ) means that each side length will reduce to be a third the length between the two figures.
10 dilation The following is a dilation. What is the scale factor? 4ft 2ft 8ft 4ft
11 The following is a dilation. Determine the scale factor of the dilation below. Is it an enlargement or reduction? (Example 3, page 20)
12 Properties of Dilations (page 18) Shape, orientation, and angles are all preserved. (this means the figure just gets larger or smaller, it doesn t turn) When finding whether a dilation has occurred, first check shape! All sides are changed by the same scale factor. To tell whether all sides have the same scale factor, determine if the points from the original to the transformed points are multiplied by the same scale factor.
13 In order to be a dilation, all side lengths must fit the scale factor. The same is true of points. If point A is located at (2,5) and point A is located at (4,10) then a dilation occurred with a scale factor of 2 (k = 2). Both the X and Y value will multiply by the SAME scale factor.
14 Is the following a dilation? Justify your answer using the properties of dilation. (Ex 1, page 19)
15 Is the following a dilation? Justify your answer using the properties of dilations. (Ex 2, page 20) In general, there cannot be a dilation if the pre-image and the transformed image share more than one point.
16 Investigating Dilations (page 32) Given a starting point and a scale factor, we can find the dilated point that matches with it. EX: If point A is located at (2,5) and the scale factor is 3 (k = 3), then A is located at 2 3, 5 3 = (6,15)
17 If we know the scale factor for a given dilation, we can find the coordinates of the dilated point that corresponds with it, by simply multiplying the x and y of the point by the scale factor. Example: If point A is at (2, 5), and the scale factor is k=3, then A is at (2x3, 5x3) = (6, 15) The same thing happens with the length of a segment. If we know the length of the pre-image segment and a scale factor, we just multiply them together to get the length of the dilated segment.
18 Example 1 (page 33)
19 Example 2 (page 34) A triangle has vertices G (2, 3), H ( 6, 2), and J (0, 4). If the triangle is dilated by a scale factor of 0.5 through center C (0, 0), what are the image vertices?
20 Example (not in WB) A triangle has vertices A (6, 5), B (2, 2), and C (-3, 4). If the triangle is dilated by a scale factor of 120% through center C (0, 0), what are the image vertices? (When given in %, you may use a calculator problems #7 & 8 on assignment)
21 Assignment 3.1 Packet: Problems # s 1-7 (separate piece of paper) WB: Page 23 # s 1-10 Page 37 # s 1-8 (separate piece of paper)
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