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Warm Up

Congruence Transformations and Coordinate Geometry. Warm Up. Lesson Presentation. Lesson Quiz. 1. Find the length of AB for A (2, 7) and B (7, –5 ). 13. ANSWER. (9, 5). ANSWER. Warm-Up. 2. What point is 6 units to the right of (3, 5) ?. 3. Are these triangles congruent?. y.

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Warm Up

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  1. Congruence Transformations and Coordinate Geometry Warm Up Lesson Presentation Lesson Quiz

  2. 1. Find the length of AB for A(2, 7) andB(7, –5). 13 ANSWER (9, 5) ANSWER Warm-Up 2. What point is 6 units to the right of (3, 5)?

  3. 3. Are these triangles congruent? y 1 –1 x Yes, by the SSS Cong. Post. or by the SAS Cong. Post. ANSWER Warm-Up

  4. ANSWER Reflection in a horizontal line Example 1 Name the type of transformation demonstrated in each picture.

  5. ANSWER Rotation about a point Example 1 Name the type of transformation demonstrated in each picture.

  6. ANSWER Translation in a straight path Example 1 Name the type of transformation demonstrated in each picture.

  7. ANSWER reflection Guided Practice Name the type of transformation shown.

  8. Example 2 Figure ABCDhas the vertices A(– 4 , 3), B(–2, 4), C(–1, 1), and D(–3, 1). Sketch ABCDand its image after the translation (x, y)→(x + 5, y –2). SOLUTION First draw ABCD. Find the translation of each vertex by adding 5 to its x-coordinate and subtracting 2 from its y-coordinate. Then draw ABCDand its image.

  9. Example 2 (x, y) → (x +5, y –2) A(–4, 3)→ (1, 1) B(–2, 4)→ (3, 2) C(–1, 1)→ (4, –1) D(–3, 1)→ (2, –1)

  10. You are drawing a pattern for a wooden sign. Use a reflection in the x-axis to draw the other half of the pattern. Woodwork Example 3 SOLUTION Multiply the y-coordinate of each vertex by –1 to find the corresponding vertex in the image.

  11. Use the vertices to draw the image. You can check your results by looking to see if each original point and its image are the same distance from the x-axis. Example 3 (x, y) → (x, –y) (–1, 0)→ (–1, 0) (–1, 2)→ (–1, –2) (1, 2)→ (1, –2) (1, 4)→ (1, –4) (5, 0)→ (5, 0)

  12. The vertices of ABCare A(1, 2), B(0, 0), and C(4, 0). A translation of ABCresults in the image DEFwith vertices D(2, 1), E(1, –1), and F(5, –1). Describe the translation in words and in coordinate notation. ANSWER Add one to each x-coordinate and subtract one from each y-coordinate, (x, y) → (x +1, y –1). Guided Practice

  13. The endpoints of RSare R(4, 5) and S(1, –3). A reflection of RSresults in the image TU, with coordinates T(4, –5) and U(1, 3). Tell which axis RSwas reflected in and write the coordinate rule for the reflection. ANSWER x-axis, (x, y) → (x, –y) Guided Practice

  14. A(–3, 1),B(–1, 3),C(1, 3),D(3, 1) mAOC = m BOD = 90° Example 4 Graph ABand CD.Tell whether CDis a rotation of ABabout the origin. If so, give the angle and direction of rotation. SOLUTION This is a 90° clockwise rotation.

  15. A(0, 1),B(1, 3),C(–1, 1),D(–3, 2) mAOC < mBOD Example 4 Graph ABand CD.Tell whether CDis a rotation of ABabout the origin. If so, give the angle and direction of rotation. SOLUTION This is not a rotation.

  16. The vertices of ABCare A(4, 4), B(6, 6), and C(7, 4). The notation (x, y)→ (x + 1, y – 3) describes the translation of ABC to DEF. Show that ABC DEFto verify that the translation is a congruence transformation. You can see that AC = DF = 3, so AC DF. S Example 5 SOLUTION

  17. Using the slopes, AB DEand AC DF. If you extend ABand DFto form G, the Corresponding Angles Postulate gives you BAC Gand A GEDF. Then,BACEDFby the Transitive Property of Congruence. S Using the Distance Formula, AB=DE=2 2 so AB DE. So, ABC DEFby the SAS Congruence Postulate. ANSWER Because ABC DEF, the translation is a congruence transformation. Example 5

  18. Tell whether PQRis a rotation of STR. If so, give the angle and direction of rotation. ANSWER yes; 180° counterclockwise Guided Practice

  19. Show that PQR STRto verify that the transformation is a congruence transformation. ANSWER PQST,PRSR, by HL, PQRSTRso it is a congruence transformation. Guided Practice

  20. ANSWER (x, y) (x – 3, y + 1) Lesson Quiz 1. Use coordinate notation to describe the translation 3 units to the left and 1 unit up.

  21. ANSWER Lesson Quiz 2. Use a reflection in the x-axis to draw the other half of the figure

  22. 3. Tell whether XY is a rotation of GH about the origin If the points are X(–6, 2), Y(–4, 3), G(6, –2), and H(4, –3). If so, give the angle and direction of rotation. ANSWER Yes; 180° clockwise or counterclockwise Lesson Quiz

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