Geometry 4.1 Translations
4.1 Warm Up Translate point P. State the coordinates of P'. 1. P(-4, 4); 2 units down, 2 units right 2. P(-3, -2); 3 units right, 3 units up 3. P(2,2); 2 units down, 2 units right 4. P(-1, 4); 3 units left, 1 unit up 5. P(2, -1); 1 unit up, 4 units right 6. P(6, 0); 4 units up, 2 units left
4.1 Essential Question How can you translate a figure in a coordinate plane?
Goals Identify and use translations in the plane. Know what a vector is and use them to solve problems.
Translation A translation is a transformation that maps every two points P and Q to points P and Q so that the following properties are true: 1. PP = QQ P 2. PP QQ P Q A Translation is an isometry. Q
Postulate 4.1 Postulate 4.1 Translation Postulate A translation is a rigid motion. Because a translation is a rigid motion, and a rigid motion preserves length and angle measure, an isometry, the following statements are true for the translation shown.
Translations in the Coordinate Plane Translation Mapping Formula: (x, y) (x + a, y + b) a and b are constants. (x + a, y + b) shifts a units horizontally. -a +a
Translations in the Coordinate Plane Translation Mapping Formula: (x, y) (x + a, y + b) a and b are constants. (x + a, y + b) shifts a units horizontally. (x + a, y + b) shifts b units vertically. -a +b +a -b
Use coordinate notation to describe the translation. a. A point is translated 10 units to the right and 8 units up. (x, y) (x + 10, y + 8) b. A point is translated 7 units to the left, and 2 units up. (x, y) (x 7, y + 2) c. A point is translated 1 unit to the left, and 2 units down. (x, y) (x 1, y 2)
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R S 3
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R S 2 3
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R (4, 4) R (4, 4) S R
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R (4, 4) S(3, 3) (3 + 3, 3 + 2) R (4, 4) S R
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R (4, 4) S(3, 3) (3 + 3, 3 + 2) S (6, 5) R (4, 4) S R
Example 1 Translate RS using: (x, y) (x + 3, y + 2) R(1, 2) (1 + 3, 2 + 2) R (4, 4) S(3, 3) (3 + 3, 3 + 2) S (6, 5) R (4, 4) S R S (6, 5)
Example 2 A Given ABC with: A(-2, 5), B(-4, 1), C(0, 0) Sketch the image after translation (x, y) (x + 4, y 4) A(-2, 5) A (2, 1) B C A B(-4, 1) B (0, -3) C(0, 0) C (4, -4) B C
Example 3 Complete the statement. If (4, 0) maps onto (5, 1), then (6, 8) will map onto. (7, 9) Why? (4 + 1, 0 + 1) (5, 1) So (6 + 1, 8 + 1) (7, 9)
Your Turn Complete the statement. If (7, 8) maps onto (4, 10), then (20, 5) will map onto. (17, 7) Why? (7 3, 8 + 2) (4, 10) So (20 3, 5 + 2) (17, 7)
Vectors A vector is a quantity that has both direction and magnitude (size). Represented with a arrow drawn between two points. Initial Point A Terminal Point B Vector AB or AB
Component form of a vector Horizontal Component S RS 6,4 R 6 4 Horizontal Component Vertical Component Vertical Component
Notation warning (2, 3) is a point on the plane. 2, 3 is a vector that can be anywhere on the plane. Any vector with a horizontal component of 2 and vertical component of 3 is the vector 2, 3. (2, 3)
Write each vector in component form. 3,1 2, 2 4,1 4,0 2, 3
Translation by Vectors From each given point, draw the vector a, b. The terminal points are where the translated points are.
Translation by Vector Translate (-4, -2) to the right 5 and up 6. We could write (x, y) (x + 5, y + 6) (-4, -2) (-4 + 5, -2 + 6) which is (1, 4).
Translation by Vector Translate (-4, -2) to the right 5 and up 6. Or we could write 5, 6 (-4, -2) + 5, 6 (1, 4).
Example 4 What vector translates (-2, 1) to (4, -3)? 6, -4 We could also say: Translate (-2, 1) by 6, -4. (-2, 1) + 6, -4 = (4, -3).
Your Turn What vector translates (4, 2) to ( 3, 1)? 7, 3
Name the vector used to translate ABCD. Name the vector used to translate ABCD. B C 6, 4 Now write the rule in coordinate notation. B C A D (x, y) (x + 6, y + 4) Give written directions for the translation. Translate ABCD to the right 6 and up 4. A D
Example 4 Translate JKL using vector -3, 3. J L K L K Notice: the vectors drawn from each point to its image are parallel. J
Example 5 Translate ABC by 4, -3. A(-4, 8) B(-10, -2) C(-22, 15) to A B C A (0, 5) B (-6, -5) C (-18, 12)
Performing Compositions Theorem 4.1 Composition Theorem The composition of two (or more) rigid motions is a rigid motion.
Example 8 Graph RS with endpoints P( 8, 5) and S( 6, 8) and its image after the composition. S Translation: (x, y) (x + 5, y 2) Translation: (x, y) (x 4, y 2) P (-8, 5) P (-3, 3) P (-7, 1) S (-6, 8) S (-1, 6) S (-5, 4) P P S P S So, the single translation rule for the composition is (x, y) (x + 1, y 4).
Summary A translation shifts a figure horizontally or vertically. A translation is an isometry. Translation formula: (x, y) (x + a, y + b) Translations can be done by vectors. Vectors have direction and magnitude. Vectors are written as h, v.
Assignment