Eighth Grade Mathematics: Translations on the Coordinate Plane
Geometry in eighth grade shifts from passive measurement of static shapes to dynamic transformational geometry, where geometric figures glide, flip, pivot, and scale across the Cartesian coordinate plane. A translation is the most fundamental of all geometric motions: a uniform slide that shifts every single point of a figure by the exact same distance in the exact same direction. In this chapter, you will master the geometric properties of translations, discover why they preserve shape and size, and learn to apply vector and coordinate shift rules.
The Nature of a Translation
A translation moves every point of a geometric figure along parallel lines of equal length. The original figure before the transformation is called the preimage, and the resulting figure after the transformation is called the image. Vertices of the image are distinguished from the preimage using prime notation, such as labeling vertex A after translation as A' (read as A prime).
Visualizing a Translation:
Preimage Triangle ABC Image Triangle A'B'C'
B B'
/ \ / / \ Slide (h units right, / / \ k units up) / A-------C ---------------------> A'------C'
Every point on triangle ABC moves along an identical path. The segment connecting A to A', the segment connecting B to B', and the segment connecting C to C' are all equal in length and completely parallel to one another.
Invariant Properties of Translations
Translations belong to a family of transformations known as rigid motions, or isometries (from the Greek words meaning equal measure). A rigid motion preserves the geometric integrity of the figure:
Preserved Properties Under Translation:
1. Distance (Side Lengths): Segment AB is congruent to Segment A'B'.
2. Angle Measures: Angle A is congruent to Angle A'.
3. Parallelism: Lines that were parallel in preimage remain parallel in image.
4. Collinearity: Points on a straight line remain on a straight line.
5. Orientation: The clockwise or counterclockwise lettering order is preserved.
Because side lengths and angle measures are strictly preserved, the translated image is always congruent to the original preimage: Triangle ABC is congruent to Triangle A'B'C'.
Coordinate Notation for Translations
On the Cartesian coordinate plane, a translation is described algebraically by adding or subtracting constant values from the x and y coordinates:
Coordinate Rule for Translation:
(x, y) ---> (x + h, y + k)
The value h represents horizontal movement:
- If h > 0, the figure shifts h units to the RIGHT.
- If h < 0, the figure shifts |h| units to the LEFT.
The value k represents vertical movement:
- If k > 0, the figure shifts k units UP.
- If k < 0, the figure shifts |k| units DOWN.
Example: Translate Triangle with vertices A(1, 2), B(3, 6), and C(5, 1)
by the rule (x, y) ---> (x - 4, y + 3).
Vertex A(1, 2): A'(1 - 4, 2 + 3) ---> A'(-3, 5)
Vertex B(3, 6): B'(3 - 4, 6 + 3) ---> B'(-1, 9)
Vertex C(5, 1): C'(5 - 4, 1 + 3) ---> C'(1, 4)
Notice that every single vertex undergoes the identical arithmetic shift: 4 subtracted from its x-coordinate and 3 added to its y-coordinate.
Chapter Exercises
Exercise 1: Triangle DEF has vertices at D(-3, 2), E(1, 5), and F(2, -1). Translate the triangle 5 units right and 4 units down. State the algebraic coordinate rule, calculate the coordinates of the image vertices D', E', and F', and explain why Triangle DEF is congruent to Triangle D'E'F'.
Exercise 2: An image vertex after translation is located at P'(7, -2). The translation rule applied was (x, y) ---> (x + 9, y - 6). Determine the original coordinates of the preimage vertex P.
Exercise 3: Explain in paragraph form why translating a line segment with a slope of 3/4 produces an image line segment that also has a slope of 3/4, referencing the coordinate translation rule.
Solutions and Explanations
Solution 1: Shifting 5 units to the right corresponds to adding 5 to the x-coordinate (h = +5), and shifting 4 units down corresponds to subtracting 4 from the y-coordinate (k = -4). The algebraic coordinate rule is (x, y) ---> (x + 5, y - 4). Applying this rule to each vertex yields: for D(-3, 2), D' = (-3 + 5, 2 - 4) = (2, -2); for E(1, 5), E' = (1 + 5, 5 - 4) = (6, 1); and for F(2, -1), F' = (2 + 5, -1 - 4) = (7, -5). Triangle DEF is congruent to Triangle D'E'F' because translations are rigid motions (isometries) that preserve all side lengths and angle measures. The distances between corresponding vertices remain identical, and the internal angles do not change, satisfying the geometric definition of congruence.
Solution 2: To find the preimage vertex P(x, y) given the image P'(7, -2) and the rule (x, y) ---> (x + 9, y - 6), set up inverse algebraic equations for each coordinate. For the horizontal coordinate: x + 9 = 7, which yields x = 7 - 9 = -2. For the vertical coordinate: y - 6 = -2, which yields y = -2 + 6 = 4. Therefore, the original preimage vertex was located at P(-2, 4). Checking: applying the rule gives (-2 + 9, 4 - 6) = (7, -2), which matches P' perfectly.
Solution 3: The slope of a line segment is defined as the vertical change divided by the horizontal change between two points: (y2 - y1) / (x2 - x1). When a segment connecting (x1, y1) and (x2, y2) is translated by the rule (x + h, y + k), the new endpoints are (x1 + h, y1 + k) and (x2 + h, y2 + k). Computing the slope of the translated image segment yields [(y2 + k) - (y1 + k)] / [(x2 + h) - (x1 + h)]. In the numerator, the added constants cancel out: y2 + k - y1 - k = y2 - y1. In the denominator, the constants also cancel out: x2 + h - x1 - h = x2 - x1. The resulting slope expression is (y2 - y1) / (x2 - x1), which is identical to the original slope. Because both the vertical rise and horizontal run are completely unaffected by the constant translation shifts h and k, the image segment retains the exact same slope of 3/4 and remains parallel to the preimage.