Unit 8: Similarity Analysis

Transcription

1 Name: Geometry Period Unit 8: Similarity Analysis Only 3 Lessons, Practice, and then QUEST In this unit you must bring the following materials with you to class every day: Please note: Calculator Pencil This Booklet A device Headphones! You may have random material checks in class Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class every day. All homework for part one of this unit is in this booklet. Answer keys will be posted as usual for each daily lesson on our website This entire booklet will be collected on the day of the quest and checked for COMPLETION You can NOT retake the quest!

2 8-1 Notes Line Dilations Today s Goal: How do we dilate lines? How does dilation impact a line that is on or off of the center of dilation? Let s Prepare! Using the word bank, a) What is preserved under a dilation, where k 1? parallelism Word Bank: distance (side length) Orientation perimeter b) What is not preserved? c) Predict it! What type of an impact will a dilation have on a line where k 1? Line Dilations on the Coordinate Plane- Together! Case 1 Case 2 a) Is the center of dilation (-2,2) on the line? a) Is the center of dilation (-2,-1) on the line? b) Did the line change after it was dilated? If so, how? BE DESCRIPTIVE! b) Did the line change after it was dilated? If so, how? BE DESCRIPTIVE! Case 3 - Special Case (center at origin): Line y = 3 x +2 is transformed by a dilation with a scale factor of 4 and centered at (0,0). The image is line 2 y = 3 x + 8. Using the above, what is the shortcut or rule can we apply to find the equation of an image after a line 2 is dilated?

3 Center is not on line Lesson Summary 1. Count horizontal and vertical distance from 2. Multiply distances by the 3. Move new distance from the Center is on line Center is the origin (0,0) The original line and the new line are to each other. The image is the equation. The original line and the new line are the. Multiply the by the and keep the same slope. The original line and the new line are to each other. Let's Try it! 1. Graph line y = 2x + 4 and its Image after a dilation with a scale factor of 1 with respect to a center of dilation at (3, 1). 3 NOTATION: D (3,1), 1 3 ) C: K: Pre-Image y-intercept Work 1. Count rise/run from C to Pre-image y-intercept 2. Multiply distance by scale factor 3. FROM C - count the new distance to the y-intercept of the image 4. The slope stays the same! Rise: x = this is new distance from C to the image. Run: x = this is new distance from C to the image. Graph new line with same slope and new y-intercept! Image

4 2. The equation of line h is 3x + y = 4. Line m is the image of line h after a dilation of scale factor 4 with center (-1, 2). What is the equation of the line m? Thinking about it backwards Line p is the image of line l. The equation of line p is given as y = 4 x + 12 If line l was dilated 3 by a scale factor of 3, centered at the origin, what is the equation of line l?

5 Practice 4. The equation of line h is 1 x y = 6. Line m is the image of 2 line h after a dilation of scale factor 2 with center at ( 1,3). What is the equation of the line m? 5. Line segment AB with endpoints (4,16) and (20,4) lies in the coordinate plane. The segment will be dilated with a scale factor of 3/4and a center at the origin to create A B. What will be the length of A B?

6 Consider the two equations: 6. Below you can see the pre-image and image of a line. Determine whether or not a dilation could have taken place. If it did, describe the dilation using details. a)pre-image: y = 4x + 16 image: y= 4x + 8 b)pre-image: y = 2x + 16 image: y -2x = 16 c) pre-image y = 2x + 4 image: y =1/2x 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?

7 8-1 Homework 1. Line y = 3x 1 is transformed by a dilation with scale factor of 2 and centered at (3,8). The line s image is. a. y = 3x -8 b. y = 3x -4 c. y = 3x -2 d. y = 3x The equation of line h is x - y = 12 Line m is the image of line h after a dilation of scale factor 1 2 origin (center is origin). What is the equation of the line m? with respect to the 3. The image of a line that is the result of a dilation of 2 with respect to the origin is shown below. What was the original line s equation (equation of the pre-image) 4. If two lines have the same equation after a dilation was performed, what can we conclude about the center of dilation with respect to that line? 5. Line y = 4x 2 is transformed by a dilation with a scale factor of 2 and centered at (3,6). The equation of the image of the line is:

8 NOW WATCH THE VIDEO ON EDPUZZLE. There will be no lesson in class on this content you are responsible for understanding it prior to our next lesson. Learning Goal: How do we describe a similarity transformation sequence to prove two triangles similar? Key fact: To determine a scale factor we recorded the following ratio: k = *Try it! Solve for the scale factor that will dilate ΔJKL to become ΔPQR. *Now let's take the numbers out! What is a way you can represent the scale factor without numbers given? Here s the plan: In this video, we will describe a similarity transformation sequence to prove triangles are similar. We will do this by showing that an image is congruent to a dilated version of a pre-image, thus making them similar! BUT FIRST What does it mean to describe? Example) Write a sequence of transformations that explains how ABC is similar to DEF The sequences of transformations that will map ABC onto DEF are ABC along so that side BC coincides with side 2. and ABC centered at by a of so that 4. Because all sides are in proportion to each other by a common the triangles are through.

9 8-2 Notes Introduction to Similarity Proofs Learning Goals: What are the minimum requirements to prove two triangles are similar? How can we use similarity to prove proportions? Check In! Apply what you learned in last night s video lesson to the question below. USING SEQUENCE OF TRANSFORMATIONS Complete the following on your own. Check on the board. Any phrase highlighted on the board SHOULD BE IN YOUR RESPONSE! Given ABC and AEF, with <CAB = <FAE and <CBA = <FEA. Describe a sequence of transformations to show that ABC ~ AEF Formal Similarity Proof Method 1 Theorem: Method 2 Theorem: Method 3 Theorem: AA~ Two pairs of corresponding angles are congruent. SSS~ All three pairs of corresponding sides are in proportion SAS~ Two pairs of corresponding sides are proportions and one pair of corresponding angles between the sides are congruent. *Tips:

10 Let s Focus on Formal Proof: Given: SQ and PR intersect at T, and QR PS. Prove PST ~ RQT Mark it Short cut plan? Statements Reasons Let s add onto this *Since PST ~ RQT we can prove the following proportion: PS = PT RQ Why? Note: After you have proven or shown that two triangles are similar, you may have another conclusion you can make. If two triangles are PROVEN similar then

11 Example # 1: What two triangles are similar, if proportion stated below is true? Example # 2: Given DE // AB. Prove that, DE BA = EC AC. Think! What are the givens? What can we infer from the given? From the diagram? MARK YOUR DIAGRAM! What are the triangles that should be SIMILAR FIRST? Statement Reason 1. DE // AB EDC 2. Alternate interior angles are congruent when lines are parallel. 3. DCE BCA DE BA = EC AC. 5.

12 Fill in the following proofs: Example # 3: Given is an isosceles triangle with base AC and BD is perpendicular to AC, prove.ab BC = AD CD. Statement Reason 1. is isosceles triangle with base AC and BD is perpendicular to AC BDA and BDC are right angles BDA BDC 4. A BCA 3. All right angles are congruent DBA and DBC are. 5. AA (two corresponding angles are congruent) AB 6. = AD CB CD 6. 4) Which triangles are always similar? Explain your choice. Explanation: (a) Equilateral triangles (b) Isosceles triangles (c) Right Triangles (d) Scalene Triangles 5) In ABC and DEF, AC = CB DF FE. Which additional information would prove ABC~ DEF?

13 6) As shown in the diagram below, circle A has a radius of 3 and circle B has a radius of 5. Using transformations, explain why Circle A is similar to Circle B. 1. circle A onto Circle B so that their centers coincide. 2. Circle A with center of dilation at and scale factor of such that circle A with circle B. Conclusion: 7) In the diagram, AB BE, DE BE and BFD ECA. Prove ABC~ DEF

14 8-3 Notes Proving Similar Triangles Day 2 Learning Goals: How can we prove two triangles are similar? How can we use similarity to prove proportions? How can we use similarity to prove cross product equivalency? Math-hoo submitted this response on his homework last night. Can you use the rubric to identify his errors and decide what score he would earn? Be sure to have a rationale for the score you think he would receive. 1. Given: WA // CH and WH and AC intersect at point T. Prove: WT = WA HT HC Let s Spark Our Thinking! *We can say that these two triangles are similar by the shortcut. *Since they are similar we can set up the following proportion: AB SR = AC We use 3 tiers to guide us through proofs of and/or involving similarity:

15 Let s try one: 1) Given is an isosceles triangle with base AC and BD is perpendicular to AC, prove AD BC = BA CD Where should we start? What tier do we need to use? What triangles are we going to prove to be similar? What s our plan? Justify at all steps! *What is given to us? *What can we infer from the givens? Statement Reason 1. is isosceles triangle with base AC and BD is perpendicular to AC BDA and BDC are right angles BDA BDC 3. All right angles are congruent. 4. A BCA BDA and BDC are

16 Perfect Practice Makes Perfect! 2) Given: AB//DE Prove: DCE ~ BCA Statement Reason Pre-proof work Do I need to re-draw my triangles separately? Did I mark my diagram? 3) Given: Trapezoid ABCD with bases BC and AD Prove: BC = EC AD EA Pre-Proof work: a) What tier type of question is this? b) Based on the given information mark your diagram appropriately (In a trapezoid the bases are ) c) Determine the 2 triangles we are looking to prove similar based on the sides we are working with (redraw the triangles below). Statement 1. Trapezoid ABCD with bases BC and AD 1. Reason 2. BC//AD CBE ADE BCE DAE 4. BCE and DAE are

17 4) Given AC BD and DE AB. Prove: AC * BD = DE * AB Where should we start? What tier do we need to use? What triangle are we going to prove to be similar? Redraw Triangles! What s our plan? Justify at all steps! *What is given to us? *What can we infer from the givens? Statement 1. AC BD and DE AB. 1. Reason 2. and are right angles B B 3. All right angles are congruent and are. 6. = AC * BD = DE * AB 7.

18 5) Given: ATX HWY, and HY//AX, prove AX HY = AT HW Hint: extend lines! Do you see any hh parallel line theorems?

19 Complete all of the following questions to help you for your quest tomorrow! Remember this quest is not retakeable! 1. In the diagram below, triangle PQR is the image of triangle XZY after a clockwise rotation of 180 and a dilation where XY = 12, YZ = 10, XZ = 7, QP = 3.5, PR = 6, RQ = 5. Which relationship must always be true? 1) m<x = 1 m<y 2 2) m<x = 2 m<y 1 3) XY = XZ PR QP 4) K = XY PR 8-4 Practice/Unit Review 2. a) Graph line y = 2x + 1 and its Image after a dilation with the following dilation: D (0, 1),3 b) What is the equation of the image?

20 3. Given: <ACB < AED. Prove: ABC~ ADE 4. Line y = 3x -1 in transformed by a dilation with a scale factor of 2 and centered at (3, 2). Write the new equation of the line's image?

21 5. Line y = 3x 1 is transformed by a dilation with scale factor of 2 and centered at (3,8). The line s image is. (1) y = 3x 8 (2) y = 3x 4 (3) y = 3x 2 (4) y = 3x 1 6. Given: ABC is isosceles with base AC; BD and EC are altitudes. Prove: BDC ~ CEA. (*Hint! What are altitudes? Does it have any other special properties?) Statements Reasons 1. ABC is isosceles with base AC; BD and EC are altitudes Isosceles Triangle Theorem Definition altitude All right angles are congruent

22 7. Given CB BA, CD DE, prove AB DE = CB CD. 8. What is an equation of the image of the line y = 3 x 4 after a dilation of a scale factor of ¾ centered at the 2 origin? 9. Explain your answer choice:

23 10. Given: Parallelogram ABCD with AB DE and BC DF Prove: AE CD = CF AD Aim for 7 statements/reasons 11. Triangle ABC and triangle ADE are graphed on the set of axes. a) Describe a transformation that maps triangle ABC onto triangle ADE b) Explain why this transformation makes triangle ADE similar to triangle A