Scale Factor of Dilation Calculator
Easily find the scale factor of a dilation using the coordinates of the center, an original point, and its image. Our scale factor of dilation calculator provides quick and accurate results.
Calculate Scale Factor
What is the Scale Factor of Dilation?
The scale factor of dilation, often denoted by 'k', is a number that describes how much larger or smaller the image of a figure is compared to the original figure after a dilation transformation. Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. It either enlarges or reduces the figure from a fixed point called the center of dilation.
If the absolute value of the scale factor |k| is greater than 1, the dilation is an enlargement. If |k| is between 0 and 1, the dilation is a reduction. If k=1, the image is congruent to the original. If k is negative, the image is on the opposite side of the center of dilation and rotated by 180 degrees compared to a positive k.
The scale factor of dilation calculator helps you find this 'k' value when you know the coordinates of the center of dilation, an original point, and its corresponding image point after dilation.
Who should use it?
Students studying geometry, teachers, engineers, architects, and anyone working with geometric transformations or scaling objects will find this scale factor of dilation calculator useful.
Common Misconceptions
A common misconception is that the scale factor must always be positive. However, a negative scale factor is possible and indicates that the image is on the opposite side of the center of dilation relative to the original point, effectively inverting it through the center.
Scale Factor of Dilation Formula and Mathematical Explanation
Let the center of dilation be C(Cx, Cy), an original point be A(Ax, Ay), and its image after dilation be A'(A'x, A'y). The scale factor 'k' is the ratio of the distance from the center to the image point (CA') to the distance from the center to the original point (CA).
Distance CA = √((Ax – Cx)² + (Ay – Cy)²)
Distance CA' = √((A'x – Cx)² + (A'y – Cy)²)
So, the scale factor k = CA' / CA, provided CA is not zero (i.e., the original point is not the center of dilation).
Alternatively, if Ax ≠ Cx, then k = (A'x – Cx) / (Ax – Cx). If Ay ≠ Cy, then k = (A'y – Cy) / (Ay – Cy). If both conditions hold, these two values of k should be the same for a uniform dilation.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| (Cx, Cy) | Coordinates of the Center of Dilation | Length units | Any real numbers |
| (Ax, Ay) | Coordinates of the Original Point | Length units | Any real numbers |
| (A'x, A'y) | Coordinates of the Image Point | Length units | Any real numbers |
| CA | Distance from Center to Original Point | Length units | ≥ 0 |
| CA' | Distance from Center to Image Point | Length units | ≥ 0 |
| k | Scale Factor | Dimensionless | Any real number (except when CA=0) |
Practical Examples (Real-World Use Cases)
Example 1: Enlargement
Suppose the center of dilation C is at (1, 1), an original point A is at (3, 3), and its image A' is at (5, 5). Using the scale factor of dilation calculator or formula:
- Cx=1, Cy=1
- Ax=3, Ay=3
- A'x=5, A'y=5
- CA = √((3-1)² + (3-1)²) = √(4+4) = √8
- CA' = √((5-1)² + (5-1)²) = √(16+16) = √32
- k = CA'/CA = √32 / √8 = √4 = 2
The scale factor is 2, indicating an enlargement.
Example 2: Reduction with Negative Scale Factor
Center C at (0, 0), original point A at (4, 2), image A' at (-2, -1).
- Cx=0, Cy=0
- Ax=4, Ay=2
- A'x=-2, A'y=-1
- CA = √((4-0)² + (2-0)²) = √(16+4) = √20
- CA' = √((-2-0)² + (-1-0)²) = √(4+1) = √5
- From x-coords: k = (-2-0)/(4-0) = -2/4 = -0.5
- From y-coords: k = (-1-0)/(2-0) = -1/2 = -0.5
The scale factor is -0.5, indicating a reduction and inversion through the center.
How to Use This Scale Factor of Dilation Calculator
- Enter Center Coordinates: Input the x (Cx) and y (Cy) coordinates of the center of dilation.
- Enter Original Point Coordinates: Input the x (Ax) and y (Ay) coordinates of a point on the original figure.
- Enter Image Point Coordinates: Input the x (A'x) and y (A'y) coordinates of the corresponding point on the image figure after dilation.
- Calculate: The calculator automatically updates, or click "Calculate". The scale factor of dilation calculator will display the scale factor 'k', distances CA and CA', and scale factors from x and y coordinates if applicable.
- Read Results: The primary result is the scale factor 'k'. Intermediate results show the distances and component-wise scale factors.
- Interpret 'k': If |k| > 1, it's an enlargement. If 0 < |k| < 1, it's a reduction. If k is negative, the image is inverted through the center.
Key Factors That Affect Scale Factor Results
- Center of Dilation: The position of the center (Cx, Cy) is crucial. All distances are measured from here.
- Original Point Coordinates: The position of (Ax, Ay) relative to the center determines the initial distance CA.
- Image Point Coordinates: The position of (A'x, A'y) relative to the center determines the final distance CA' and thus the scale factor.
- Relative Positions: The alignment of C, A, and A' determines if k is positive or negative. If A' is on the same ray from C as A, k is positive. If on the opposite ray, k is negative.
- Zero Distance: If the original point is the center of dilation (CA=0), the scale factor is undefined unless the image point is also the center (in which case it can be any k, or it's just a point fixed). Our scale factor of dilation calculator handles this.
- Consistency: For a uniform dilation, the scale factor calculated from x-coordinates and y-coordinates should be the same.
Frequently Asked Questions (FAQ)
- What if the scale factor is 1?
- If k=1, the image is congruent to the original figure (no change in size), and it's the same orientation if k=1 (or inverted if k=-1 with |k|=1, but that's usually considered k=-1).
- What does a negative scale factor mean?
- A negative scale factor (e.g., k=-2) means the image is on the opposite side of the center of dilation compared to the original, and its size is scaled by |k| (e.g., |k|=2, so twice as large).
- Can the scale factor be zero?
- If k=0, the image of every point is the center of dilation itself. The entire figure collapses to the center point C.
- What if the original point is the center of dilation?
- If A=C, then CA=0. The scale factor k=CA'/CA is undefined unless A'=C as well (CA'=0). If A=C and A'=C, k can be any value as 0*k=0. The center of dilation is a fixed point.
- How is dilation related to similarity?
- Dilation is a similarity transformation. The original figure and its image after dilation are always similar.
- Does the scale factor have units?
- No, the scale factor is a dimensionless ratio of two lengths.
- Can I find the image coordinates if I know k, C, and A?
- Yes. A'x = Cx + k * (Ax – Cx) and A'y = Cy + k * (Ay – Cy). You might need an image point calculator for that.
- What if I have lengths instead of coordinates?
- If you know the length of a segment in the original figure and the corresponding segment in the image, the scale factor is Image Length / Original Length. Our scale factor from lengths calculator can help.