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

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

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

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

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

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

Guided Practice Name the type of transformation shown. reflection ANSWER

Example 2 Figure ABCD has the vertices A(– 4 , 3), B(–2, 4), C(–1, 1), and D(–3, 1). Sketch ABCD and 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 ABCD and its image.

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)

Example 3 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 SOLUTION Multiply the y-coordinate of each vertex by –1 to find the corresponding vertex in the image.

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) 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.

Guided Practice The vertices of ABC are A(1, 2), B(0, 0), and C(4, 0). A translation of ABC results in the image DEF with 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 The endpoints of RS are R(4, 5) and S(1, –3). A reflection of RS results in the image TU , with coordinates T(4, –5) and U(1, 3). Tell which axis RS was reflected in and write the coordinate rule for the reflection. ANSWER x-axis, (x, y) → (x, –y)

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

Example 4 Graph AB and CD. Tell whether CD is a rotation of AB about the origin. If so, give the angle and direction of rotation. A(0, 1), B(1, 3), C(–1, 1), D(–3, 2) SOLUTION m AOC < m BOD This is not a rotation.

Example 5 The vertices of ABC are 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 DEF to verify that the translation is a congruence transformation. SOLUTION S You can see that AC = DF = 3, so AC DF.

Example 5 A Using the slopes, AB DE and AC DF . If you extend AB and DF to form G, the Corresponding Angles Postulate gives you BAC G and G EDF. Then, BAC EDF by the Transitive Property of Congruence. S Using the Distance Formula, AB = DE = 2 2 so AB DE . So, ABC DEF by the SAS Congruence Postulate. Because ABC DEF, the translation is a congruence transformation. ANSWER

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

Guided Practice Show that PQR STR to verify that the transformation is a congruence transformation. ANSWER PQ  ST, PR  SR, by HL, PQR  STR so it is a congruence transformation.

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

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

Lesson Quiz 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