L11 Dilations and Similarity 11.1 Ratio Review Warmup Per Date. 1. Fill in the table below as demonstrated in the sample top row.

Transcription

1 11.1 Ratio Review Warmup Per Date 1. Fill in the table below as demonstrated in the sample top row. Ratio Fraction Equation Written x to y is 3 x y =3 x = 3y x is 3 times as big as y x to y is ½ x y =2 x = 1 4 y x is 5 times as big as y x to y is Find the ratios of the pairs of sides of the triangle. AB BC = AB AC = AC BC = AC AB = Page 1

2 11.1 Ratio Review Warmup Per Date 3. Determine whether each pair of figures are rigid motion transformations of each other. If not, explain which property fails. A B C D E F Page 2

3 11.2 Grandma s Gazebo Per Date This lesson investigates the final type of transformation, known as a dilation, which is NOT a rigid motion transformation. 1. Below is a bird s eye view of Grandma s Gazebo, as depicted on the architect s blueprints. Because of her poor eyesight, Grandma is having difficultly seeing the details of the Gazebo drawing. Help her out by drawing a new version that is twice as big, by following the instructions below. F A E G B D C i. From the center, extend the dotted line segment GAto be the ray GA. Repeat for the ray GB. ii. Measure the length of the line segments GAand GB. (They should be equal, and in fact also equal to AB, since ΔABG is equilateral.) iii. Mark the points A' and B ' on the rays GA and GB, respectively, so that the line segments GA' and GB' are twice as long as line segments GAand GB. (Use a ruler or compass.) iv. Now measure the length of the line segment A 'B'. a. What is the relationship between the length of A'B' and the length of AB?!##"!### " b. How are the lines AB and A' B' related? (Note: not the line segments, but the lines.) Page 3

4 11.2 Grandma s Gazebo Per Date Imagine repeating Steps 3 and 4 at each vertex. What would be the shape of the resulting figure? c. Try it out. (Repeat Steps 3 and 4 at each vertex) What do you notice? d. Use patty paper or a protractor to compare the angles ABC and A B C, and the angles ABG and A B G. What do you notice? e. Measure the distance from the center G to the midpoint of AB. What do you think the distance is from the center G to the midpoint of AB? ' ' Try it out. f. Repeat this comparison for the line segments connecting the midpoints of AB and CD, and those connecting the midpoints of AB ' ' and CD. ' ' What do you notice? g. What is the ratio of the corresponding sides between the double-scaled drawing and the original? The ratio is: This means that when we extend the rays from a specific point of the gazebo by a multiple of, then the side lengths multiply by a factor of. h. What is the ratio of the corresponding angles between the double-scaled drawing and the original? The ratio is: This means that when we extend the rays from a specific point of the gazebo by a multiple of, then the angles. When we increase the size of the gazebo, the length measurements, but the angle measurements. Page 4

5 11.2 Grandma s Gazebo Per Date i. What do you think the ratio of corresponding sides would be if we had created a triplescale diagram? j. What do you think the ratio of corresponding angles would be if we had created a triplescale diagram? Summary: This dilation transformation seems to have preserved the, but increased the. Page 5

6 11.2 Grandma s Gazebo Per Date 2. Grandma s eyes are pretty weak. Use the process we just completed earlier to draw a copy of the hexagon gazebo that is three times larger than the original picture. F A E G B D C Be sure to check the ratio of the diagonals of your new diagram to the original one. What is this ratio? Reflection: Do you think there is anything special about increasing the size by an integer multiple (2, 3, 5, etc.)? For example, suppose we wanted to make the gazebo drawing 5.3 times as large. Do you think the same results would hold? Page 6

7 11.2 Grandma s Gazebo Per Date 3. Suppose we took a hexagon with side lengths of 3 inches (i.e. your results from the previous problem) and placed it on the ground so that its center was at the precise center of where Grandma will locate her gazebo. Grandma s Gazebo has side lengths of 8 feet. Calculate how many times larger the actual gazebo is than the diagram placed on the ground, by filling in the following blanks. Gazebo drawing side length: inches Gazebo actual side length: feet = inches How many times larger: a. In order to build a hexagonal gazebo with 8 ft sides, we would need to increase the size of our diagram by a factor of, which is what we call the size ratio between the two hexagons. b. Measure the radius of your 3-inch sided Gazebo from Problem 2, it should also be 3. Therefore, the radius of the actual Gazebo should be. c. If we wanted to increase our current drawing (of side length 3 inches) to a larger drawing with 1 foot sides, we would need to increase the size of our diagram by a factor of, the size ratio between the two hexagons. This ratio is referred to as the scale factor (#VOC), which is a non-negative number that we use to increase or decrease the size of our image, while maintaining the same shape. 4. Assume you are the contractor and you need to lay out the Gazebo with 8 ft. sides. You know where the center is located, but you need to locate each of the vertices. You can assume you have a very large compass, since a compass can be emulated with a string attached to a stake in the ground (e.g. located at the center of the Gazebo), and that you have a 25 ft. tape measure, which is essentially a long ruler. What steps would you follow to lay out the Gazebo? Note: you may choose your first vertex arbitrarily, as long as its distance from the center is correct. Page 7

8 11.2 Grandma s Gazebo Per Date 5. Get up and go: With your teacher and classmates, enlarge or compress some figures in your classroom to measure scale factor. Use the Elmo and projector, or find like objects, to measure side lengths. Original object s side lengths:,,, Projected object s side lengths:,,, Ratio between lengths:,,, Therefore scale factor is: Page 8

9 11.3 Practice Set 1 Per Date 1. Practice computing the size ratios between related objects (in mm). Assume you dilate the figure on the left to get the figure on the right. Find each scale factor. Scale Factor: Scale Factor: Scale Factor: Scale Factor: Page 9

10 11.4 Homework 1 Per Date 2. Pair up the following triangles based on their common shapes. A B C D E F Pair 1: & Pair 2: & Pair 3: & Scale Factor: Scale Factor: Scale Factor: Measure each side length and angle. Determine if your pairing makes sense based on the measurements you derived. Explain how the measurements verify each pair is related in the same manner as were your earlier pairs of Gazebos. Page 10

11 11.5 Dilation Per Date The type of transformation that we used to enlarge Grandma s Gazebo, where we expanded our figure from its center, is an example of a special type of non-rigid transformation that we refer to as dilation, which we will formally define below. Let s try to dilate figures from points other than a center. 1. Below is Δ ABC. Dilate this triangle by a scale factor of 2 from vertex A and label the new triangle Δ AB ' C '. Here point A is playing the same role as point G in the Gazebo exercise. Follow the same steps used there (i.e. draw a ray with endpoint A through point B, etc.) B A C Check that your work is accurate. Measure each side length of ΔABC and ΔAB ' C ' to determine if the ratio of their lengths confirms our scale factor of 2. Show your work below. Remember to label your measurements with the side lengths they represent. Are the angles of the new triangle the same as the corresponding angles of the original triangle? When dilating a triangle using one of its vertices as the center of dilation, how many individual image points completely determine the entire image? Page 11

12 11.5 Dilation Per Date 2. Scale Factor Review: A. How does the image of a diagram compare to the original diagram when it is dilated using a scale factor of 2? B. How does the image of a diagram compare to the original diagram when it is dilated using a scale factor of 1? C. How does the image of a diagram compare to the original diagram when it is dilated using a scale factor of ½? D. How does the image of a diagram compare to the original diagram when it is dilated using a scale factor of 1.5? 3. Now that you have practiced the skill of dilation from a vertex, practice this non-rigid transformation on the triangles below with non-integer scale factors (using patty paper and a straightedge). You may choose whichever vertex you like from which to dilate. A. ΔEFG dilated to ΔEF ' G ' with scale factor of 1.5 E G F Page 12

13 11.5 Dilation Per Date B. ΔKLM dilated to ΔK' L' M with a scale factor of 0.5 K L M C. ΔQRS dilated to ΔQ ' RS ' with scale factor 2.5 R S Q Page 13

14 11.5 Dilation Per Date 4. How would two images compare for problem 3A above if the first resulted from using point E as the center of dilation, and the second resulted from using point F as its center of dilation? We now move to the most general type of dilation, where the point of dilation is outside the figure. Follow the directions below to dilate Δ ABC from point D with a scale factor of 2. D B C A i. Draw rays DA and DB. ii. Measure the lengths DAand DB. iii. Mark the point A' and B' on the rays DA and DB, respectively, so that the line segments DA' and DB ' are twice as long as line segments DAand DB. (In other words, points A' and B'are twice as far from D as A and B, respectively.) How are the lengths AB and ABrelated? ' '!##"!### "!##"!#### "!##"!#### " How are the lines AB and A' B' related? What about AC and A'C ', BC and B'C '. iv. Repeat this process for point C to form triangle A B C. v. Use patty paper or a protractor to compare corresponding angles in the two triangles. What do you notice about the corresponding angles in the two triangles? Page 14

15 11.5 Dilation Per Date You have now seen three types of dilation: 1) from the center of the figure (Grandma s Gazebo), 2) from a vertex of a figure, and 3) from a point outside a figure. A dilation with center D and scale factor r (#VOC) is a transformation that takes each point P (P DP ' not equal to D) to a point P along the ray DP such that the ratio of the distances r DP =. (Note: Unless the scale factor is 1, a dilation is a non-rigid transformation) Guided Notes: A scale factor is a number. A dilation from point D with scale factor r takes a line segment of length L to a line segment of length rl g. In other words, if the image under this dilation of line segment AB is the line segment AB, ' ' then the of AB ' ' is r-times the length of.!##" Furthermore, the lines AB!### " and A' B' are. As a result, the image of a dilation of a polygon will result in a polygon that has the same but the length of the will be by a factor of r. A dilation with center D and scale factor r takes a line not containing D to a. A dilation with center D and scale factor r takes a line containing D to. A dilation with center D and a scale factor r takes an angle formed by rays AB and AC to a. In order to dilate a triangle, using one of the vertices as the center of dilation, you only need to locate the image for in order to completely determine the dilated image. Page 15

16 11.6 Introduction to Similarity Per Date A fundamental question in is Does Figure A have the same shape as Figure B? What we have seen is that the image of a dilation has the same shape as its pre-image. Two such objects, the image and the pre-image, are said to be similar. Before formalizing the meaning of similarity, let s intuitively investigate the connection between this important concept and congruence. 1. The following pairs of objects intuitively appear to be similar in the sense that they appear to have the same shape. Use patty paper or a protractor to check the congruency of their corresponding angles. Page 16

17 11.6 Introduction to Similarity Per Date 2. One way to determine if a pair of figures is similar is to attempt to match up the angles. However, since the size of the figures differ, you can only match up one angle at a time. Go back and cut out the right figure in each of the three previous pairs. Lay your cutout approximately where it was previously located. Slide it around, as if you were applying a rigid motion transformation until one of the vertices and its angle matches. Note: you may also need to flip your cutout over if a reflection is required. Now that an angle is matched up, does there appear to be a dilation that would transform the smaller figure into the larger figure? What is the approximate scale factor for each pair? Since dilations preserve angle measures, this proves the two figures have the same angles. 3. Let s investigate the connection between same shape and congruence with a final example. a. Translate triangle ABC below until vertex A coincides with vertex D. Use patty paper or cut out triangle ABC. b. Rotate this image of triangle ABC an appropriate amount clockwise until line segment AC is horizontal (as is line segment DF). c. Reflect the smaller triangle image over the vertical line passing through point D. B A E D F C It should now be clear that angles A and D are congruent, and further, if you were to dilate the smaller triangle by an appropriate scale factor it would coincide precisely with triangle DEF, which shows the two triangles have congruent angles. In conclusion, a similarity transformation (#VOC) is a rigid motion transformation (e.g. the translation, rotation, and reflection used above) followed by a dilation. Similarity transformations demonstrate the connection between objects that have the same shape, but are not congruent (i.e. they can be made congruent via an appropriate dilation). Figure 1 and Figure 2 are said to be similar (#VOC) if Figure 2 is the image of Figure 1 under a similarity transformation (e.g. triangle DEF coincided with the image of the similarity Page 17

18 11.6 Introduction to Similarity Per Date transformation of triangle ABC, so the two triangles are similar, which simply means they have congruent corresponding angles). The notation we use is ΔABC : ΔDEF. 4. Two boats leave a dock at the same time. Boat A travels 10 miles per hour directly south and Boat B travels 30 miles per hour directly East (See diagram below.) At each moment in time the three points corresponding to the dock and the positions of the two points form a triangle. Is the triangle formed after 10 minutes of travel similar to the triangle formed after 30 minutes of travel? Explain the reasoning you used to answer this question to a classmate. Dock Boat B 10 min Boat B 30 min Boat A 10 min Boat A 30 min 5. A boy is running from second base to third base. At each moment in time during his run the three points corresponding to his position, third base, and home plate, form a triangle. Locate the points corresponding to his position a quarter of the way and halfway to third base, and draw the triangles. Are the two triangles similar? Explain your answer to a classmate. 2 nd Base 3 rd Base Home Plate Page 18

19 11.7 Exit Pass Per Date Rotate ΔBCD 90 CW about point A. Dilate the image by a scale factor of 2. Label it Δ BCD ' ' ' C B D A Explain how you knew which vertices to label B, C, and D. The image and pre-image are related, in that ΔBCD and ΔB' CDare ' '. Page 19

20 11.8 Homework 2 Per Date Complete the following rigid motion transformation and dilations to create new images of the given figures. 1. Rotate ΔEFG 90 CCW about point A and dilate the image by a scale factor of 1.5, and label the resulting image Δ E' F' G'. E F G A Check your work by measuring the corresponding angles: m E = m F = m G = m E = m F = m G = The image and pre-image are related, in that ΔEFG and ΔE' FG ' ' are. Page 20

21 11.8 Homework 2 Per Date 2. Reflect ΔHIJ about line m, translate the resulting image using the translation defined by T(x, y) = (x, y 5), and dilate the resulting image by a scale factor of 2 and label it Δ H' I' J'. H m I J The image and pre-image are related, in that ΔHIJ and ΔH' I' J ' are. Page 21