Midpoints, reflections and rotations
Calculate the image of a point or figure using a stated line or centre, and check invariant distances.
What you will tackle
- Find coordinate midpoints
- Reflect in x = c, y = c and y = x
- Rotate a quarter-turn about the origin
- Rotate a half-turn about an arbitrary centre
Midpoints and reflections
The midpoint of (x₁,y₁) and (x₂,y₂) averages corresponding coordinates. Reflection in x = c sends (x,y) to (2c − x,y); reflection in y = c sends it to (x,2c − y). Reflection in y = x swaps the coordinates. Points on the mirror line stay fixed.
State the direction and centre
A 90° counterclockwise rotation about the origin maps (x,y) to (−y,x); clockwise maps it to (y,−x). A 180° rotation about centre (h,k) maps it to (2h − x,2k − y). Reflections and rotations preserve distances, while coordinate signs alone do not specify the centre.
| Vertex | x | y |
|---|---|---|
| A | 1 | 1 |
| B | 4 | 1 |
| C | 1 | 3 |
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Find the midpoint of (−2,1) and (4,9). Reflect (3,−1) in x = 1, in y = 2, and in y = x.
Show a starting hint
Check that midpoint-to-endpoint distances match.
Compare with the explained answer
- The midpoint is ((−2 + 4)/2,(1 + 9)/2) = (1,5).
- For x = 1, the image is (−1,−1); for y = 2, it is (3,5).
- Swapping coordinates for y = x gives (−1,3).
Answer / response criteria: (1,5); (−1,−1); (3,5); (−1,3).
Independent practice
Independent 1
Rotate (2,−3) by 90° counterclockwise about the origin, then by 180° about (1,2), each starting from the original point. Reflect the displayed triangle's vertices A(1,1), B(4,1), C(1,3) in y = x.
Compare with the explained answer
- The quarter-turn maps (2,−3) to (3,2).
- The half-turn maps the original point to (2 − 2,4 + 3) = (0,7).
- The reflected triangle is A′(1,1), B′(1,4), C′(3,1), preserving its side lengths.
Answer / response criteria: (3,2); (0,7); reflected vertices as stated.
Application practice
Application 1
A camera marker at (7,−1) moves to its image after a half-turn about (2,3). Find that image, show the centre is their midpoint, and explain why a reflection in x = 2 gives a different result.
Compare with the explained answer
- The half-turn image is (4 − 7,6 + 1) = (−3,7).
- The midpoint of (7,−1) and (−3,7) is (2,3).
- Reflection in x = 2 gives (−3,−1), preserving the y-coordinate rather than reversing displacement in both directions.
Answer / response criteria: Half-turn image (−3,7); midpoint (2,3); mirror image (−3,−1).
What can you explain now?
A camera marker at (7,−1) moves to its image after a half-turn about (2,3). Find that image, show the centre is their midpoint, and explain why a reflection in x = 2 gives a different result.
Compare with the explanation
Half-turn image (−3,7); midpoint (2,3); mirror image (−3,−1).
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide