SHSAT: Transformations and similar figures
Apply translations, reflections, rotations and dilations to coordinates, and scale perimeters and areas of similar figures.
How do you find where a point lands after a transformation, and how do dilations change measurements?
Short answer
Use the coordinate rule for each transformation, in the order given. Rigid motions keep sizes the same; a dilation by k multiplies lengths by k and areas by k².
Know first: Plotting points in all four quadrants, Squaring fractions
Step 1 of 5
Each transformation has a coordinate rule
A transformation moves every point of a figure by the same rule. Translations, reflections and rotations are rigid motions: they keep lengths, angles and area the same. A dilation keeps angles but multiplies every length by the scale factor.
For coordinate questions, apply the rule to each point. When there are two steps, do them in the order given.
| Transformation | (x, y) becomes |
|---|---|
| translate a right and b up | (x + a, y + b); left and down use minus |
| reflect across the x-axis | (x, −y) |
| reflect across the y-axis | (−x, y) |
| rotate 90° counterclockwise about the origin | (−y, x) |
| rotate 90° clockwise about the origin | (y, −x) |
| rotate 180° about the origin | (−x, −y) |
| dilate by k about the origin | (kx, ky) |
For a center other than the origin, measure from the center, apply the rule, then add the center back.
Step 2 of 5
Apply the steps in order
Worked example: Reflect, then rotate
The point (3, −2) is reflected across the y-axis, then rotated 90° counterclockwise about the origin. What is the final point?
- A(−3, −2)
- B(−2, 3)
- C(2, −3)
- D(2, 3)
- 1
Reflect across the y-axis
(x, y) becomes (−x, y): (3, −2) becomes (−3, −2).
- 2
Rotate 90° counterclockwise
(x, y) becomes (−y, x): (−3, −2) becomes (2, −3).
- 3
Check the quadrant
(−3, −2) is in quadrant III. A quarter turn counterclockwise moves it to quadrant IV, where (2, −3) is.
- 4
See why the other choices are there
(−3, −2) stops after the reflection. (−2, 3) rotates clockwise. (2, 3) rotates the starting point and skips the reflection.
Answer
(2, −3)
Wrong: Dilate (5, 1) by a factor of 3 about the center (2, −1): (3 × 5, 3 × 1) = (15, 3).
Right: Measure from the center: (5 − 2, 1 − (−1)) = (3, 2). Multiply by 3: (9, 6). Add the center back: (2 + 9, −1 + 6) = (11, 5).
Multiplying the coordinates directly works only when the center is the origin.
Step 3 of 5
Use the rule, then check the picture
Check yourself · Question 1
The point (−5, 2) is translated 4 units left and 3 units up. What is its image?
Answer: A
Left subtracts from x: −5 − 4 = −9. Up adds to y: 2 + 3 = 5. The image is (−9, 5).
Check yourself · Question 2
The point (1, 4) is rotated 180° about the point (3, 1). What is its image?
Answer: B
From the center (3, 1), the point is 2 left and 3 up. A half-turn sends it 2 right and 3 down from the center: (3 + 2, 1 − 3) = (5, −2).
Trap answer: (−1, −4)
Using the origin as the center
- Why it looks right
- The rule (x, y) becomes (−x, −y) is the one you memorized for 180° rotations.
- Why it's wrong
- That rule works only about the origin. Here the center is (3, 1), so measure from it.
- The right answer
- (5, −2): 2 right and 3 down from (3, 1).
Check yourself · Question 3
Two similar triangles have corresponding sides of 4 and 10. The smaller triangle has an area of 12 square units. What is the area of the larger triangle?
Answer: C
The length scale factor is 10/4 = 5/2. Areas scale by (5/2)² = 25/4. 12 × 25/4 = 75.
Common mistake
Mixing up the clockwise and counterclockwise rotation rules.
- Why it's tempting
- The two rules look alike: both swap x and y and change one sign.
- Do this instead
- Check the quadrant. A counterclockwise quarter turn moves a point from quadrant I to quadrant II, so (1, 2) should become (−2, 1).
Step 4 of 5
What to remember
Remember
- Apply each coordinate rule in the order given, and check the image's quadrant.
- For a center other than the origin, measure from the center, transform, then add the center back.
- Rigid motions keep lengths and area. A dilation by k multiplies lengths by k and area by k².
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which coordinate changed when the point moved right, which changed when it moved down, and in which direction?