Size Transformations in the Coordinate Plane

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Karin Cain
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1 Size Transformations in the Coordinate Plane I.e. Dilations (adapted from Core Plus Math, Course 2) Concepts Lesson Objectives In this investigation you will use coordinate methods to discover several characteristics of size transformations including: The effects of size transformations on length, area, and angle measure (extended to parallel and perpendicular lines) for various figures. The relation of the center and magnitude of a size transformation to pairs of preimage/image points. Modeling Size Transformations In the previous investigation, you found patterns in the coordinates of preimage/image pairs for transformations with which you were familiar. For those transformations, the distance between any pair of preimage points was the same as the distance between their images. As a result, under these rigid transformation, a polygon and its image had the same size and shape--they were congruent. In this investigation, you will reverse the procedure. You will start with a rule relating coordinates of any preimage and its image, and you will explore how the transformation affects familiar shapes. As you complete the problems in this investigation, look for an answer to this question: How can coordinates be used to rescale or resize a shape? 1. Consider the transformation defined by the following rule: Preimage Image (x, y) (3x, y) This rule is read the x-coordinate of the image is 3 times the x-coordinate of the preimage; the y-coordinate of the image is the same as the y-coordinate of the preimage. a. Which of figures II, III, or IV appears to be the image of Figure I under this transformation? Explain your reasoning. b. On the coordinate grid, plot the points X(1,1), Y(5,1), and Z(5,5). Draw XYZ and its image under this transformation. c. Examine your preimage and image shapes. What characteristics of XYZ are also characteristics of its image?

2 d. How do you think the perimeter of XYZ will compare to the perimeter of its image? Test your conjecture. e. How do you think the area of XYZ will compare to the area of its image? Test your conjecture. f. Which of Figures II, III, or IV could be the image of Figure I when transformed by the rule: (x, y) (x, 3y)? What clue(s) did you use? 2. A size transformation (or dilation) of magnitude 3 centered at the origin is defined by the following rule: Preimage Image (x, y) (3x, 3y) a. What is the scale factor of this dilation? b. On the diagram, draw the image of quadrilateral ABCD under this size transformation. Label image vertices A, B, C, and D. c. Examine your preimage and image shapes. Make a list of all of the properties of quadrilateral ABCD that seem to also be properties of quadrilateral A B C D. d. Describe how the two shapes seem to differ. 3. Making visual comparisons, as you did in Problem 2, is useful; but such comparisons should be made with some skepticism. You should always seek additional evidence to support or refute your visual conjectures. This is where coordinate representations and formulas for distance and slope can be very helpful. Use these ideas to examine more carefully quadrilateral ABCD and its image quadrilateral A B C D that you drew in Problem 2. a. Compare the length of AB with the length of AB. Does the same relation hold for other preimage/image pairs of segments? Explain. b. How does AB appear to be related to AD ( hint: find their slopes)? Does the same relationship hold for their images? Give evidence to support your claim.

3 c. How do the perimeters of quadrilateral ABCD and A B C D compare? d. How does BC appear to be related to AD? Is this relationship true for their images? Justify your conclusion. e. What kind of quadrilateral is ABCD? Is the image quadrilateral A B C D the same kind of quadrilateral? Explain your reasoning? f. How do the areas of quadrilaterals ABCD and A B C D appear to be related? State a conjecture. Test your conjecture. How does the magnitude of the size transformation come into play here? 4. Your drawing of quadrilateral ABCD and its image quadrilateral A B C D under the size transformation of magnitude 3 should look like the drawing to the right.. a. Use a ruler to draw lines through A and A, B and B, C and C, and D and D. Extend the lines to intersect the x and y axes. What do you notice about the intersection of these four lines? b. Consider your observation from part a. How do you think you could identify the center of a dilation given the preimage and image figures? c. Find the equations of the four lines in part a. Use these equations to verify your observation in Part a. i. Equation of line : AA ii. Equation of line : BB iii. Equation of line : CC iv. Equation of line : DD d. The size transformation has its center at the origin since the lines in Part a intersect at (0, 0). What is the image of the center (0, 0) under this size transformation?

4 e. Compare the distances from the center O to a point and to the image of that point. State a conjecture. i. Find the distances from O to A and O to A : O to B and O to B : O to C and O to C : O to D and O to D : ii. Do these distances confirm your conjecture? f. Generalize your finding. i. How should the distances from O(0, 0) to P(a,b) and from O(0, 0) to P (3a, 3b) be related? ii. Show why this must be the case by calculating the distances OP and OP. g. Complete the following statement: If O is the center of a size transformation with magnitude k and the image of P is P, then OP = and OP OP =. Properties of Size Transformations 5. Next, consider a size transformation with magnitude 0.5 and center at the origin. a. Write a rule for this size transformation: b. On the diagram shown, plot and label the image of quadrilateral PQRS under the size transformation. How do you think quadrilateral PQRS and its image are related in terms of shape? In terms of size?

5 c. Compare segment lengths in the image with corresponding lengths in quadrilateral PQRS by filling in the following table: Preimage Length Image Length PQ QR RS SP P Q QR RS SP. i. How does the magnitude 0.5 affect the relation between lengths? ii. How does the magnitude 0.5 affect the relation between perimeters? d. Find the area of the image quadrilateral. Compare it to the area of quadrilateral PQRS. How does the magnitude 0.5 affect the relation between areas? 6. Now consider PQR with vertices P(3, 4), Q(-3, 2), and R(-2, -1). a. Sketch PQR on the coordinate grid. b. Sketch the image P QR, resulting from transforming PQR with size transformation of magnitude 2.5 and center at the origin. c. Compare lengths of corresponding preimage and image sides. Preimage Length Image Length PQ QR RP P Q QR RP

6 d. How are PQ and QR related? Give evidence to support your claim. e. How are P Q and QR related? Give evidence to support your claim. f. Use the information in Parts d and e to help you determine the area of PQR and P QR. Compare the areas and relate them to the magnitude Draw XY Z with vertices X(3, 2), Y(1, -3), and Z(-4, -4), and find images of XY Z when transformed with magnitudes 1.5 and 3. a. In each case, compare sides and areas of the preimage and image triangles. Are the results of your comparisons consistent with what you would have predicted? Explain.

7 Summarize the Mathematics In this investigation you explored size transformations and their properties. 1. Explain why the transformation in problem 1, (x, y) (3x, y) is or is not a size transformation. 2. Suppose a size transformation with magnitude k > 0 and center at the origin O maps A onto A, B onto B, and C onto C. a. Write a rule that can be used to obtain the image of any point (x, y) in the coordinate plane under this size transformation. State your rule in words AND in symbolic form. b. How is the length of A B related to the length of AB? c. If ABC has an area of 25 square units, what is the area of ABC? Why does this make sense in terms of the formula for the area of a triangle?how is the distance from O to C related to the distance from O to C? d. Where do and intersect? Does intersect there too? AA CC BB 3. How are size transformations similar to rigid transformations? How are they different? 4. Complete the statement. Rigid transformations produce figures, and size transformations (dilations) produce figures. 5. Are any rigid transformations also size transformations? If so, what is the magnitude (scale factor) of a size transformation that is also a rigid transformation?

Unit 7. Transformations

Unit 7. Transformations Unit 7 Transformations 1 A transformation moves or changes a figure in some way to produce a new figure called an. Another name for the original figure is the. Recall that a translation moves every point