Unit 7 Scale Drawings and Dilations

Transcription

1 Unit 7 Scale Drawings and Dilations Day Classwork Day Homework Friday 12/1 Unit 6 Test Monday 12/4 Tuesday 12/5 Properties of Scale Drawings Scale Drawings Using Constructions Dilations and Scale Drawings with Various Centers of Dilation 1 HW HW 7.2 Wednesday 12/6 Thursday 12/7 Friday 12/8 Triangle Side Splitter Theorem Midsegments of Triangles Unit 7 Quiz 1 3 HW 7.3 Dividing a Line Segment into Equal Segments 4 HW 7.4 Dilations in the Coordinate Plane 5 HW 7.5 Monday 12/11 Review Unit 7 Quiz 2 6 Review Sheet Tuesday 12/12 Wednesday 12/13 Review 7 Review Sheet Unit 7 Test 8

2 SCALE DRAWINGS The scale factor r is the ratio of any length in a scale drawing relative to its corresponding length in the original figure. A scale factor r > 1 results in an enlargement of the original figure. A scale factor of 0 < r < 1 results in a reduction of the original figure. Examples 1. Use construction tools to create a scale drawing of ABC with a scale factor of r =2. Steps: Measure the length of BC and B C. What do you notice? Measure the angles B, C, B ', and C'. What do you notice?

3 2. Use construction tools to create a scale drawing of DEF with a scale factor of r = 3. What properties does your scale drawing share with the original figure? Explain how you know. 3. Use construction tools to create a scale drawing of XYZ with a scale factor of 1 r. 2

4 1 4. Use construction tools to create a scale drawing of PQR with a scale factor or r. What 4 properties do the scale drawing and the original figure share? Explain how you know. 5. EFG is provided below, and one angle of scale drawing E' F ' G' is also provided. Use construction tools to complete the scale drawing so that the scale factor is r 3. What properties do the scale drawing and original figure share? Explain how you know.

5 6. Triangle ABC is provided below, and one side of scale drawing A' B' C ' is also provided. Use construction tools to complete the scale drawing and determine the scale factor. MORE SCALE DRAWINGS For r>0, a dilation with center O and scale factor r is denoted D Or, For the center O, D Or, (O)=O For any other point P, D Or, (P) is the point P on the ray OP so that OP ' r OP 1. Create a scale drawing of the figure below about center O and scale factor 1 r. 2 Does A B C D E look like a scale drawing? How can we verify this?

6 2. Create a scale drawing of the figure below about center O and scale factor r 2. Verify that the resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and the corresponding angles are equal in measurement. 3. Create a scale drawing of the figure below about center O and scale factor r 3.

7 4. A' B' C ' is a scale drawing of ABC. Use your straight edge to determine the location of the center O used for the scale drawing Use the figure below, center O, a scale factor of r to create a scale drawing. Verify that the 2 resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and that the corresponding angles are equal in measurement.

8 DILATIONS Dilation Theorem Example Figure If a dilation with center O and scale factor r sends point P to P and Q to Q, then P Q =r PQ. Furthermore, if r 1 and O,, and Q are the vertices of a triangle, then PQ P 'Q'. Examples: 1. Given the diagram below, determine if DEF is a scale drawing of DGH. Explain why or why not. 2. Two different points R and Y are dilated from S with a scale factor of 3, and RY = 15. Use the 4 Dilation Theorem to describe two facts that are known about R Y.

9 Summary a. Which transformations of the plane are distance-preserving transformations? Provide an example of what this property means. b. Which transformations of the plane preserve angle measure? Provide one example of what this property means. c. Which transformation is not considered a rigid motion and why?

10 TRIANGLE PROPORTIONS Triangle Proportionality Theorem (Triangle Side Splitter Theorem) Example Figure A line segment divides two sides of a triangle into segments of proportional lengths if and only if it is parallel to the third side of the triangle. Examples In PRQ, ST RQ. Find the following measures based on the given information. 1. PT = 7.5, TQ = 3, and SR = 2.5. Find PS. 2. PS = 12.5, SR = 5, and PT = 15. Find TQ. 3. PS 12, ST 8, and SR 6. Find RQ. 4. In DEF, EH = 3, HF = 9, and DG is one third the length of GF. Is DE GH?

11 A is a segment with endpoints that are the midpoints of two sides of the triangle. Every triangle has three midsegments. Triangle Mid-Segment Theorem Example Figure A midsegment of a triangle is parallel to one side of the triangle, and its length is one half the length of that side. Examples In the figure, XY and XZ are midsegments of 1. XZ RST. Find each measure below. 2. ST 3. m RYX 4. x = 5. x = Perimeter of ABC = y = Another special case of the Triangle Proportionality Theorem involves three of more parallel lines cut by two transversals.

12 6. Megan is drawing a hallway in one-point perspective. She uses the guidelines shown to draw two windows on the left wall. If segments AD, BC, WZ, and XY are all parallel, AB = 8 cm, DC = 9 cm, and ZY = 5 cm, find WX. 7. In RST, the midpoints of each side have been marked by points X, Y, and Z. Mark the halves of each side divided by the midpoint with a congruency mark. Remember to distinguish congruency marks for each side. Draw mid-segments XY, YZ, and XZ. Mark each mid-segment with the appropriate congruency mark from the sides of the triangle. a. What conclusion can you draw about the four triangles within RST? Explain Why. b. State the appropriate correspondences between the four triangles within RST. c. State a correspondence between RST and any one of the four small triangles.

13 DIVIDING A LINE SEGMENT (DILATION METHOD) 1. Divide AB into four segments of equal lengths. Describe your steps: 2. Use the Dilation Method to divide PQ into 9 equal-sized pieces.

14 3. If the segment below represents the interval from zero to one on the number line, locate and label 4 7.

15 DILATIONS IN THE COORDINATE PLANE A dilation is a transformation of an object by increasing or decreasing the object by a factor of r. Enlargement Types of Dilations Symbols Examples Figures Reduction Examples Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. a. b. c. d.

16 1. Quadrilateral JKLM has vertices J(-2, 4), K(-2, -2), L(-4, -2), M(-4, 2). Graph the image of JKLM after a dilation centered at the origin with a scale factor of Find the image of each polygon below with the given vertices after a dilation centered at the origin with the scale factor. a. Q (0, 6), R (-6, -3), S (6, -3); r = 1/3 b. A (2, 1), B (0, 3), C (-1, 2), D (0, 1); r = 2

17 3. A dilation with center O1 and scale factor 1/2 maps figure F onto F. A dilation with center O2 and scale factor 1/2 maps figure F to F. Draw figures F and F, and then find the center O and scale factor r of the dilation that takes F to F If a figure T is dilated from center O1 with a scale factor r 1 to yield image T, and figure T is 4 4 then dilated from center O2 with a scale factor r2 to yield figure T. Explain why figure T and T 3 are congruent.