Eighth Grade Mathematics: Dilations, Scale Factors, and Center of Dilation
While translations, reflections, and rotations move figures without altering their size, real-world applications often require resizing shapes while maintaining their exact geometric proportions. Architects draw blueprints, cartographers create maps, and graphic designers scale digital vector artwork. The transformation responsible for resizing figures proportionally is a dilation. In this chapter, you will master the mechanics of dilations, investigate the role of the scale factor and center of dilation, and discover why dilations preserve shape while changing size.
What is a Dilation?
A dilation is a transformation that produces an image that has the exact same shape as the preimage, but a different size. Every dilation is characterized by two essential parameters:
1. The Center of Dilation: A fixed reference point from which all points are expanded or contracted
(in eighth grade, this is typically the origin (0, 0)).
2. The Scale Factor (k): The constant multiplier that determines how much the figure grows or shrinks.
Visualizing Dilation from the Center of Dilation (O):
B' (Distance from O is k * OB)
/ B / / \ / / \ / A-----C A'-------C'
/
O (Center of Dilation)
Every line connecting the center of dilation O to an image point P' passes directly through the corresponding preimage point P. The distance from the center to the image point is equal to the scale factor k multiplied by the distance from the center to the preimage point: OP' = k * OP.
The Magnitude of the Scale Factor
The scale factor, symbolized by the lowercase letter k, governs whether the dilation is an enlargement or a reduction:
Scale Factor Conditions (k > 0):
- If k > 1: Enlargement (Image is LARGER than preimage, moves further from center)
- If 0 < k < 1: Reduction (Image is SMALLER than preimage, moves closer to center)
- If k = 1: Identity (Image is congruent to preimage, size does not change)
Examples of Scale Factors:
k = 3 ---> Enlargement: Every length is multiplied by 3 (tripled).
k = 1/2 ---> Reduction: Every length is multiplied by 1/2 (halved).
k = 0.75 ---> Reduction: Every length is 75% of original length.
Notice that in eighth grade, scale factors are strictly positive. A negative sign in advanced geometry represents a dilation combined with a 180-degree rotation.
Coordinate Rule for Dilations Centered at the Origin
When the center of dilation is the origin (0, 0), the algebraic coordinate rule is remarkably simple: you multiply both the x-coordinate and the y-coordinate by the scale factor k:
Coordinate Rule (Center at (0, 0)):
(x, y) ---> (k * x, k * y)
Example 1 (Enlargement with k = 2):
Vertex A(2, 3) ---> A'(2*2, 2*3) = A'(4, 6)
Vertex B(5, 1) ---> B'(2*5, 2*1) = B'(10, 2)
Example 2 (Reduction with k = 1/3):
Vertex P(6, 9) ---> P'((1/3)*6, (1/3)*9) = P'(2, 3)
Vertex Q(-3, 12) ---> Q'((1/3)*(-3), (1/3)*12) = Q'(-1, 4)
What Changes and What Remains the Same Under Dilation?
Because a dilation alters the size of a figure, it is NOT a rigid motion (isometry). However, it possesses critical geometric invariants:
Properties PRESERVED Under Dilation:
1. Angle Measures: Every angle in the image is CONGRUENT to its corresponding angle in preimage!
2. Parallelism: Parallel lines remain parallel; image segments are parallel to preimage segments.
3. Collinearity: Straight lines remain straight lines.
4. Orientation: The circular order of vertices remains unchanged.
Properties CHANGED Under Dilation:
1. Side Lengths: Side lengths are NOT preserved; every side length is multiplied by k.
2. Perimeter: Perimeter of image = k * (Perimeter of preimage).
3. Area: Area of image = (k^2) * (Area of preimage).
Notice that while side lengths scale by k, area scales by k squared! If you double the dimensions of a rectangle (k = 2), its area quadruples (2^2 = 4).
Chapter Exercises
Exercise 1: Triangle ABC has vertices at A(-2, 4), B(4, 6), and C(2, -2). Dilate Triangle ABC with a center of dilation at the origin and a scale factor of k = 1/2. State whether this is an enlargement or reduction, and calculate the coordinates of the image vertices A', B', and C'.
Exercise 2: A rectangular photograph with a width of 4 inches and a length of 6 inches is enlarged using a dilation with a scale factor of k = 3.5. Calculate the dimensions, perimeter, and area of both the original photograph and the enlarged photograph, explaining the relationship between the scale factor and the area change.
Exercise 3: Explain in paragraph form why the slope of a line segment connecting two points on a coordinate plane does not change when the segment undergoes a dilation centered at the origin, referencing the dilation coordinate rule and the slope formula.
Solutions and Explanations
Solution 1: Because the scale factor is k = 1/2, which satisfies 0 < k < 1, this dilation is a reduction: the image will be smaller than the preimage and located closer to the origin. The coordinate rule for a dilation centered at the origin with k = 1/2 is (x, y) ---> ((1/2)x, (1/2)y). Applying this rule to each vertex yields: for A(-2, 4), A' = ((1/2)(-2), (1/2)4) = (-1, 2); for B(4, 6), B' = ((1/2)4, (1/2)6) = (2, 3); and for C(2, -2), C' = ((1/2)2, (1/2)(-2)) = (1, -1). The image vertices are A'(-1, 2), B'(2, 3), and C'(1, -1).
Solution 2: For the original photograph, the width is 4 inches and the length is 6 inches. The original perimeter is 2(4) + 2(6) = 20 inches, and the original area is 4 * 6 = 24 square inches. Under a dilation with scale factor k = 3.5, the new dimensions are: width = 4 * 3.5 = 14 inches, and length = 6 * 3.5 = 21 inches. The new perimeter is 2(14) + 2(21) = 28 + 42 = 70 inches. Notice that 20 * 3.5 = 70, showing the perimeter scaled by k. The new area is 14 * 21 = 294 square inches. To understand the relationship between the scale factor and area: Area_new = (k * width) * (k * length) = k^2 * (width * length) = k^2 * Area_orig. Here, k^2 = 3.5^2 = 12.25. Multiplying the original area of 24 by 12.25 gives exactly 24 * 12.25 = 294 square inches. The area scales by the square of the scale factor.
Solution 3: The slope of a line segment between two points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). When the segment undergoes a dilation centered at the origin with scale factor k, the coordinates of the endpoints transform to (kx1, ky1) and (kx2, ky2). Computing the slope of the dilated image segment yields m_image = (ky2 - ky1) / (kx2 - kx1). Factoring the common multiplier k out of both the numerator and the denominator gives m_image = [k * (y2 - y1)] / [k * (x2 - x1)]. Because k is non-zero, the factor k in the numerator and denominator cancels out completely (k/k = 1), leaving m_image = (y2 - y1) / (x2 - x1). Because the slope of the image segment is identically equal to the slope of the preimage segment, dilations preserve the slope of line segments and map lines to parallel lines.