Translations
Give the vector notation for the translation that moves point (2, −1) to (5, 3).
⟨3,4⟩
State the center and angle of rotation that sends the point (1,0) to (0,1) about the origin.
Center: origin; angle: 90° counterclockwise.
What is the image of point (4, −2) after reflection across the x-axis?
(4,−2)↦(4,2)
Define dilation in terms of center and scale factor and state whether it preserves angles.
Dilation: transformation with a center and scale factor k multiplying distances from the center by k; it preserves angle measures.
What is the result of performing a translation by ⟨2,0⟩ followed by a translation by ⟨−2,3⟩? Give the single equivalent translation vector.
Equivalent translation: ⟨0,3⟩
Describe how coordinates change under a translation by vector ⟨a,b⟩
(x, y)↦(x + a, y + b)
Give the coordinates of (3, 4) after a 90° counterclockwise rotation about the origin.
(3,4)↦(−4,3)
Reflect point (−3, 5) across the line y=x; give the coordinates.
(−3,5)↦(5,−3).
Dilate point (2, 3) with center at origin and scale factor 3; give the image.
(2,3)↦(6,9).
If you reflect a figure across the x-axis then across the y-axis, what single transformation is equivalent? Describe and justify.
Reflection across x then y = rotation 180° about the origin (point reflection through origin).
Translate triangle with vertices (0,0), (2,1), (1,3) by ⟨−3,2⟩ and give the new coordinates
Add ⟨−3,2⟩: (0,0)→(−3,2), (2,1)→(−1,3), (1,3)→(−2,5)
Rotate triangle with vertices (1,2), (3,2), (1,5) by 180° about the origin; list the image vertices.
180° about origin: (1,2)→(−1,−2), (3,2)→(−3,−2), (1,5)→(−1,−5)
A trapezoid is reflected across a vertical line. Which properties must be preserved? (Name at least three.)
Preserved: side lengths, angles, parallelism; orientation is reversed; congruence preserved.
A dilation centered at (1,1) with scale factor 2 sends point (3,4) to which coordinates? Show your calculation.
Vector from (1,1) to (3,4) is (2,3); scaled → (4,6); image = (1+4, 1+6) = (5,7).
Apply the following to point (1,1): rotate 90° counterclockwise about the origin, then translate by ⟨2,−1⟩. Give final coordinates.
(1,1) 90° CCW → (−1,1); then translate by ⟨2,−1⟩ → (1,0).
A translation maps segment AB to A'B'. What properties of the segment are preserved under any translation? (List at least three.)
Preserved: length, angle measures, parallelism, orientation, betweenness (congruence)
Describe how to perform a rotation of 270° clockwise about the origin as a rotation of another standard degree measure (give degree and direction)
270° clockwise is equivalent to 90° counterclockwise
Describe the reflection that maps triangle ABC to A'B'C' where A(1,1) maps to A'(−1,1). Give the line of reflection.
Since A maps (1,1)→(−1,1), the line of reflection is x=0 (the y-axis).
How does a dilation with factor k affect the length of a segment? Provide the formula relating original and image lengths.
Lengths scale by k: L′=kL.
Determine whether the composition "dilation by factor 2 about the origin" followed by "rotation 180° about the origin" is commutative with performing the rotation first and then the dilation. Explain why or why not.
They commute when both centered at origin because both are linear maps (matrices) about the same center; order does not matter. If centers differ, order matters.
Given a line with equation y=2x+1, find the equation of its image after translation by vector ⟨−4,5⟩.
Translate: substitute (x+4,y−5) into original to find image: (y−5)=2(x+4)+1 → y=2x+14
A point P is at (2, −1). Find the image of point P after it's rotated 90∘ clockwise about the origin
(-1, -2)
Reflect the point (2, 3) across the line y=−x. Then show algebraically that reflection is an isometry by verifying distance from origin is preserved.
Reflection across y=−x maps (2,3)↦(−3,−2). Distances from origin: both √13, so isometry verified.
Given triangle with side lengths 5, 12, 13. After a dilation with center at origin and scale factor 3/4, what are the new side lengths? Is the new triangle similar to the original? Explain.
New side lengths: 15/4, 9, 39/4. Yes — triangles are similar.
Given transformation T defined as: reflect across the line y=x, then translate by ⟨3,−2⟩, then dilate by factor 1/2 about the origin. Apply T to point (4,2) and show each step's result.
Step 1 (reflect across y=x): (4,2) → (2,4). Step 2 (translate by ⟨3,−2⟩): (2,4) → (5,2). Step 3 (dilate by 1/2 about origin): (5,2) → (5/2,1).