Geometry: Transformations (High School)
Translations, reflections, rotations, dilations, and compositions on the coordinate plane
Geometry: Transformations (High School)
Translations, reflections, rotations, dilations, and compositions on the coordinate plane
Geometry - Grade 9-12
- 1
Triangle ABC has vertices A(2, 1), B(5, 1), and C(3, 4). Translate the triangle by the vector < -3, 2 >. Give the coordinates of A', B', and C'.
Add -3 to each x-coordinate and add 2 to each y-coordinate.
The translated coordinates are A'(-1, 3), B'(2, 3), and C'(0, 6). Each original point moves 3 units left and 2 units up. - 2
Point P(-4, 7) is reflected across the x-axis. What are the coordinates of P'?
The coordinates of P' are (-4, -7). A reflection across the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate. - 3
Point Q(6, -2) is reflected across the y-axis. What are the coordinates of Q'?
Across the y-axis, left and right switch, but height does not change.
The coordinates of Q' are (-6, -2). A reflection across the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same. - 4
Point R(3, -5) is rotated 90 degrees counterclockwise about the origin. What are the coordinates of R'?
The coordinates of R' are (5, 3). A 90 degree counterclockwise rotation about the origin maps (x, y) to (-y, x). - 5
Point S(-2, 8) is rotated 180 degrees about the origin. What are the coordinates of S'?
A 180 degree rotation sends a point to the opposite quadrant through the origin.
The coordinates of S' are (2, -8). A 180 degree rotation about the origin maps (x, y) to (-x, -y). - 6
Triangle DEF has vertices D(1, 2), E(4, 2), and F(2, 6). Dilate the triangle by a scale factor of 3 centered at the origin. Give the coordinates of D', E', and F'.
The dilated coordinates are D'(3, 6), E'(12, 6), and F'(6, 18). A dilation centered at the origin with scale factor 3 multiplies each coordinate by 3. - 7
Rectangle WXYZ has vertices W(-1, 1), X(3, 1), Y(3, 4), and Z(-1, 4). Reflect the rectangle across the line y = x. Give the coordinates of W', X', Y', and Z'.
Switch the x-coordinate and y-coordinate of each point.
The reflected coordinates are W'(1, -1), X'(1, 3), Y'(4, 3), and Z'(4, -1). A reflection across y = x maps each point (x, y) to (y, x). - 8
A transformation maps every point (x, y) to (x + 5, y - 1). Name the transformation and describe it in words.
The transformation is a translation. It moves every point 5 units to the right and 1 unit down. - 9
A transformation maps every point (x, y) to (-x, y). Name the transformation and identify the line of reflection.
Only the left-right position changes.
The transformation is a reflection across the y-axis. It changes the sign of each x-coordinate while keeping each y-coordinate the same. - 10
Point A(2, 3) is first translated by <4, -5> and then reflected across the y-axis. What are the final coordinates of A''?
Perform the transformations in the order given.
After the translation, A' is (6, -2). Reflecting across the y-axis gives A''(-6, -2). - 11
Point B(-3, 4) is reflected across the x-axis and then rotated 90 degrees clockwise about the origin. What are the final coordinates of B''?
After reflection across the x-axis, B' is (-3, -4). A 90 degree clockwise rotation maps (x, y) to (y, -x), so B'' is (-4, 3). - 12
A triangle has side lengths 5 cm, 7 cm, and 9 cm. It is dilated by a scale factor of 0.5. What are the side lengths of the image triangle, and is the image congruent or similar to the original?
Multiply each side length by the scale factor.
The image side lengths are 2.5 cm, 3.5 cm, and 4.5 cm. The image is similar to the original, but it is not congruent because the side lengths changed. - 13
Quadrilateral JKLM is rotated 270 degrees counterclockwise about the origin. Which single rotation would produce the same result, and in what direction?
A 90 degree clockwise rotation about the origin would produce the same result. Rotating 270 degrees counterclockwise is equivalent to rotating 90 degrees clockwise. - 14
Line segment AB has endpoints A(-2, 5) and B(4, 1). It is reflected across the x-axis. Does the length of the segment change? Explain.
Rigid motions preserve length and angle measure.
The length of the segment does not change. A reflection is a rigid motion, so it preserves distances and produces a congruent image. - 15
Triangle PQR has vertices P(0, 0), Q(6, 0), and R(2, 4). Triangle P'Q'R' has vertices P'(0, 0), Q'(3, 0), and R'(1, 2). Describe the transformation from triangle PQR to triangle P'Q'R'.
Compare corresponding coordinates such as Q(6, 0) and Q'(3, 0).
The transformation is a dilation centered at the origin with scale factor 1/2. Each coordinate of the original triangle is multiplied by 1/2 to create the image.