Geometry: Transformations
Translations, reflections, rotations, dilations, and composition of transformations
Geometry: Transformations
Translations, reflections, rotations, dilations, and composition of transformations
Math - Grade 9-12
- 1
Triangle ABC has vertices A(2, 1), B(5, 1), and C(3, 4). Translate the triangle 4 units left and 3 units up. Give the coordinates of A', B', and C'.
Subtract 4 from each x-coordinate and add 3 to each y-coordinate.
The translation rule is (x, y) to (x - 4, y + 3). The image coordinates are A'(-2, 4), B'(1, 4), and C'(-1, 7). - 2
Point P(-6, 4) is reflected across the y-axis. What are the coordinates of P'?
A reflection across the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same. Therefore, P' is (6, 4). - 3
Point M(3, -7) is reflected across the x-axis. What are the coordinates of M'?
For an x-axis reflection, use the rule (x, y) to (x, -y).
A reflection across the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate. Therefore, M' is (3, 7). - 4
Quadrilateral WXYZ has vertices W(-2, 1), X(1, 3), Y(4, 0), and Z(0, -2). Rotate the quadrilateral 90 degrees counterclockwise about the origin. Give the coordinates of W', X', Y', and Z'.
Switch the coordinates, then make the new x-coordinate the opposite of the original y-coordinate.
A 90 degree counterclockwise rotation about the origin uses the rule (x, y) to (-y, x). The image coordinates are W'(-1, -2), X'(-3, 1), Y'(0, 4), and Z'(2, 0). - 5
Point R(-5, 2) is rotated 180 degrees about the origin. What are the coordinates of R'?
A 180 degree rotation about the origin changes the sign of both coordinates. Therefore, R' is (5, -2). - 6
Triangle DEF has vertices D(1, 2), E(4, 2), and F(2, 6). Dilate the triangle about the origin by a scale factor of 3. Give the coordinates of D', E', and F'.
Multiply every x-coordinate and every y-coordinate by 3.
A dilation about the origin with scale factor 3 multiplies each coordinate by 3. The image coordinates are D'(3, 6), E'(12, 6), and F'(6, 18). - 7
A figure is transformed by the rule (x, y) to (x + 7, y - 2). Name the transformation and describe it in words.
The transformation is a translation. It moves every point 7 units to the right and 2 units down. - 8
A point (x, y) is transformed to (-x, y). Name the transformation and identify the line of reflection.
Compare how the x-coordinate and y-coordinate change.
The transformation is a reflection across the y-axis because the x-coordinate changes sign while the y-coordinate stays the same. - 9
Segment AB has endpoints A(-1, 5) and B(3, 2). It is translated to A'(4, 1) and B'(8, -2). What translation vector was used?
From A to A', the change is 5 units right and 4 units down. From B to B', the same change occurs. The translation vector is <5, -4>. - 10
Triangle JKL has vertices J(2, -1), K(6, -1), and L(4, -4). First reflect the triangle across the x-axis, then translate it 3 units left. Give the final coordinates of J'', K'', and L''.
Perform the transformations in the order given.
After reflecting across the x-axis, the coordinates are J'(2, 1), K'(6, 1), and L'(4, 4). Translating 3 units left gives J''(-1, 1), K''(3, 1), and L''(1, 4). - 11
A triangle has side lengths 4, 7, and 8. It is dilated by a scale factor of 1/2. What are the side lengths of the image triangle?
A dilation multiplies all lengths by the scale factor. The image side lengths are 2, 3.5, and 4. - 12
Rectangle ABCD has vertices A(0, 0), B(6, 0), C(6, 4), and D(0, 4). It is dilated about the origin by a scale factor of 2. What is the perimeter of the image rectangle?
Perimeter is multiplied by the same scale factor as side lengths.
The original rectangle has side lengths 6 and 4, so its perimeter is 20. A dilation by a scale factor of 2 doubles all lengths, so the image perimeter is 40 units. - 13
A figure is rotated 90 degrees clockwise about the origin. Write the coordinate rule for this transformation and use it to find the image of Q(-3, 8).
For a 90 degree clockwise rotation, the original y-coordinate becomes the new x-coordinate.
A 90 degree clockwise rotation about the origin uses the rule (x, y) to (y, -x). For Q(-3, 8), the image is Q'(8, 3). - 14
Two triangles are congruent. Triangle A is mapped onto Triangle B by a single transformation. The orientation is reversed, and corresponding points are the same distance from the y-axis on opposite sides. What transformation maps Triangle A to Triangle B?
The transformation is a reflection across the y-axis. A reflection reverses orientation and places corresponding points the same distance from the line of reflection. - 15
Point S(4, -2) is transformed by the rule (x, y) to (2x - 1, 2y + 3). Find S' and describe the transformation as a dilation followed by a translation.
Apply the multiplication part first, then the addition or subtraction part.
Substituting S(4, -2) gives S'(2(4) - 1, 2(-2) + 3), so S'(7, -1). This can be described as a dilation about the origin by a scale factor of 2 followed by a translation 1 unit left and 3 units up.