Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:A geometric transformation changes a figure by applying the same mapping rule to all its points.
Common stuck point:The procedure for geometric transformation is the easy part; the trap is moving only one vertex. Asking "What rule sends each original point to its new point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: What rule sends each original point to its new point?
Worked Examples
Example 1
easy
Name and describe the four basic geometric transformations.
Answer
Translation, Rotation, Reflection, Dilation.
First step
1
Step 1: Translation — slides every point by the same vector (a,b): (x,y)→(x+a,y+b).
Full solution
2
Step 2: Rotation — turns every point around a fixed center by a given angle.
3
Step 3: Reflection — flips every point over a line (the axis of reflection).
4
Step 4: Dilation — scales every point away from or toward a center by a scale factor k.
Translation, rotation, and reflection are isometries — they preserve distances and angles (rigid motions). Dilation changes size but preserves shape (similarity transformation). Together these transformations form the foundation of geometric transformation theory.
Example 2
medium
Point P(3,−1) is reflected over the y-axis, then translated by (2,5). Find the final image.Reflect P over the y-axis, then translate by ⟨2, 5⟩
Example 3
medium
Rotate the point (2,3) by 180° about the origin.180° rotation of (2,3) about the origin
Example 4
medium
Apply (x,y)↦(−y,x) to (3,4), and identify the transformation.Rule (x,y) → (−y, x) applied to (3,4): 90° CCW rotation
Example 5
medium
Rotate (4,1) by 90° CCW, then translate by ⟨2,−3⟩. Find the final image.Rotate (4,1) by 90° CCW → (−1,4), then translate by ⟨2,−3⟩ → (1,1)
Example 6
hard
What is the image of (5,2) under a reflection across the line y=x, followed by a 90° CCW rotation about the origin?Reflect (5,2) across y=x → (2,5), then rotate 90° CCW → (−5,2)
Example 7
hard
Rotate (3,4) by 90° CCW about the point (1,2).Rotate (3,4) by 90° CCW about the point (1,2)
Example 8
challenge
A point (2,3) is rotated 60° CCW about the origin. Give its image (round to two decimals).60° CCW rotation of (2,3) about the origin
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Which geometric transformations are isometries (preserve distances)?
Example 2
hard
Show that a 180° rotation about the origin is equivalent to the transformation (x,y)→(−x,−y). Verify with the point (3,4).180° rotation of (3, 4) about the origin maps to (−3, −4)
Example 3
easy
Name the four basic geometric transformations.
Example 4
easy
Which transformation changes a figure's size: translation or dilation?
Example 5
easy
A transformation maps (x,y)→(x+2,y−3). What type is it?
Example 6
easy
When you apply a transformation to a figure, how many of its points move?
Example 7
easy
Which three transformations are 'rigid' (preserve size and shape)?
Example 8
easy
A figure is flipped over a line. What transformation is this?
Example 9
easy
The original figure is called the 'preimage'. What is the result of a transformation called?
Example 10
easy
A transformation maps (x,y)→(2x,2y). Does it change the figure's size?
Example 11
medium
Which transformations preserve orientation, and which reverses it?
Example 12
medium
A figure is translated, then rotated. Is the final image congruent to the original?
Example 13
medium
A figure is dilated by factor 3. Is the image congruent or similar to the original?
Example 14
medium
Apply (x,y)→(−y,x) to the point (3,2). What transformation is this, and what's the image?Rule (x,y) → (−y, x) rotates 90° CCW: (3,2) → (−2,3)
Example 15
medium
Why does tracking only the vertices of a polygon suffice to transform the whole figure?
Example 16
medium
A transformation maps (x,y)→(x,−y). Name it.
Example 17
medium
Two transformations done in sequence form a 'composition'. If a slide is followed by a flip, is the result still a single transformation?
Example 18
medium
A dilation has scale factor 21. Does it enlarge or shrink the figure?
Example 19
challenge
A figure is reflected across the y-axis, then across the x-axis. What single transformation is equivalent?
Example 20
challenge
A triangle with area 12 is dilated by factor 3, then translated. Find the image's area.
Example 21
challenge
Which property is preserved by ALL four basic transformations (translation, rotation, reflection, dilation)?
Example 22
challenge
Explain why every rigid motion can be built from at most three reflections.
Example 23
easy
Apply (x,y)↦(x+4,y−2) to the point (3,5).Translation (x,y) → (x+4, y−2): find the image of (3,5)
Example 24
easy
Reflect (4,9) across the y-axis.Reflection of (4,9) across the y-axis
Example 25
easy
Rotate (5,0) by 90° counterclockwise about the origin.90° CCW rotation of (5,0) about the origin
Example 26
easy
Translate (0,0) by the vector ⟨−3,7⟩.
Example 27
medium
Apply (x,y)↦(y,x) to (4,7). Which transformation is this?Reflection of (4,7) across the line y = x
Example 28
medium
Apply (x,y)↦(y,−x) to (3,4), and identify the transformation.Rule (x,y) → (y, −x) applied to (3,4): 90° CW rotation
Example 29
medium
Compose two reflections across parallel lines ℓ1:x=0 and ℓ2:x=3. What is the result?
Example 30
hard
A triangle has vertices (0,0), (4,0), (0,3). Dilate from origin by factor 2, then translate by ⟨1,1⟩. Find the new vertices.Dilate triangle by factor 2 from origin, then translate by ⟨1,1⟩
Example 31
hard
A triangle has area 12 and perimeter 20. After dilation by factor 5, what are the new area and perimeter?
Example 32
hard
Two figures have areas 9 and 36 and are similar. What is the ratio of their corresponding sides?
Example 33
challenge
Prove that the composition of any rotation and a translation is again a rotation (possibly about a different center).