8th Grade Dilations and Similar Figures Lesson (Free PDF)
This dilations similarity lesson for 8th Grade covers dilations and similar figures (Common Core 8.G.A.3, 8.G.A.4), with a free PDF to print.
A dilation enlarges or reduces a figure by a scale factor k from a center point; from the origin, each coordinate is multiplied by k. The image has equal angles but lengths multiplied by k, so it is similar. Two figures are similar if a sequence of rotations, reflections, translations and dilations maps one onto the other.
Grade 8 · Common Core State Standards: 8.G.A.3, 8.G.A.4
Key facts in this dilations similarity lesson
- Dilation from the origin: \((x, y) \to (kx, ky)\).
- \(k > 1\) enlarges; \(0 < k < 1\) reduces; \(k = 1\) leaves the figure unchanged.
- Angles stay equal; side lengths multiply by \(k\); area multiplies by \(k^{2}\).
- Similar figures have equal angles and proportional sides.
- Use \(\frac{\text{new}}{\text{old}} = k\) to find missing lengths.
What is a dilation?
A dilation resizes a figure from a fixed point called the center. The scale factor \(k\) tells how much the figure grows or shrinks. If \(k > 1\), it is an enlargement; if \(0 < k < 1\), it is a reduction. When the center is the origin, \((x, y) \to (kx, ky)\).
To check a dilation, draw a dashed ray from the center through each vertex. The image vertex lies on that ray, at \(k\) times the distance from the center. For the triangle with vertex \((4, 2)\) and scale factor \(\frac{1}{2}\), the image vertex is at \((2, 1)\), halfway from the origin. Reductions bring the figure closer to the center and enlargements push it farther away. The center itself never moves.
What are similar figures?
Two figures are similar if one can be mapped onto the other by a sequence of rotations, reflections, translations and dilations. Matching angles are equal and matching sides are proportional.
The triangle with sides 3, 4, 5 and the triangle with sides 6, 8, 10 are similar with scale factor 2: \(\frac{6}{3} = \frac{8}{4} = \frac{10}{5} = 2\).
How do you use similar figures to find a missing length?
- Match the corresponding sides.
- Write a proportion: \(\frac{\text{new}}{\text{old}} = k\).
- Solve for the missing length.
Worked example. A 6-foot person casts a 4-foot shadow while a tree casts a 10-foot shadow. The sun makes similar triangles, so \(\frac{h}{6} = \frac{10}{4}\). Then \(h = 6 \cdot 2.5 = 15\) feet.
To describe the sequence of moves between two similar figures, name the dilation and the rigid motions. For example, a triangle with vertices \((1, 1)\), \((2, 1)\), \((1, 3)\) maps to \((-2, -2)\), \((-4, -2)\), \((-2, -6)\) by a dilation with scale factor 2 followed by a \(180^{\circ}\) rotation about the origin.
Dilations similarity lesson: questions and answers
What is a scale factor?
The scale factor is the number each length is multiplied by in a dilation. It equals the ratio of an image length to the matching original length.
Is a dilated figure congruent to the original?
Only when the scale factor is 1. Otherwise the side lengths differ, so the figures are similar but not congruent.
Do angles change in a dilation?
No. Dilations preserve angle measures and parallel lines. Only lengths (and areas) change.
How does area change when the scale factor is k?
Area is multiplied by \(k^{2}\). A scale factor of 3 makes the area 9 times as large.
How do I know two shapes are similar?
Show equal corresponding angles and proportional corresponding sides, or describe a sequence of moves with a dilation that maps one onto the other.
How to use this dilations similarity lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the dilations similarity lesson as a PDF with the button below and keep it with your notes.
All Grade 8 chapters: Grade 8 math.
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