A dilation has scale factor 3. A preimage side is 8. Find the image side.
Unit 05 · Lesson 07 · Similarity, Dilations & Proportional Reasoning
Indirect Measurement, Shadows & Mirrors
Use similar triangles in shadows, mirrors, and line-of-sight situations.
Learning intention
Use accurate diagrams, definitions, measurements, and reasoning to use similar triangles in shadows, mirrors, and line-of-sight situations.
Success evidence
- Label the figure and units before calculating.
- Select and justify the governing definition, theorem, relationship, or construction.
- Show enough work that another learner can audit the conclusion.
- Check the result against the diagram, scale, and stated constraints.
Visual model
Use dilations and proportional relationships to compare figures, construct scale models, and measure inaccessible quantities.
k=\frac{\text{image length}}{\text{preimage length}}\frac{a}{d}=\frac{b}{e}=\frac{c}{f}AA,\ SAS\sim,\ SSS\sim
Worked reasoning
Process: Establish similarity first, write a proportion with matching sides, solve, and check that the magnitude agrees with the scale factor.
- Read the figure and rewrite all given information with correct notation.
- Identify the exact relationship that connects the known and unknown quantities.
- Substitute, construct, or prove one justified step at a time.
- State the result with units and a sentence explaining what it means.
Misconception check: Corresponding side ratios must be written in a consistent order; mixing image-to-preimage and preimage-to-image reverses the scale factor.
Khaemenes Academy · Open-Age Geometry
Indirect Measurement, Shadows & Mirrors
Find, construct, prove, or explain each result. Show reasoning and label units.
Name
Date
Triangles are similar. A side 7 corresponds to 7; a side 7 corresponds to x. Find x.
Which angle evidence alone is sufficient for triangle similarity?
How does a dilation with scale factor 1/2 affect angle measure?
A 1:50 drawing shows a wall 8 cm long. What is the actual length?
Why should scale units be written explicitly?