K Khaemenes GeometryOpen-Age · Printable Visual Lesson
Unit 05 Course Home Student Lesson

Unit 05 · Lesson 05 · Similarity, Dilations & Proportional Reasoning

Side-Splitter and Proportional Segments · Teacher Key

Use dilations and proportional relationships to compare figures, construct scale models, and measure inaccessible quantities.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intention

Use accurate diagrams, definitions, measurements, and reasoning to use dilations and proportional relationships to compare figures, construct scale models, and measure inaccessible quantities.

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

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image

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.

  1. Read the figure and rewrite all given information with correct notation.
  2. Identify the exact relationship that connects the known and unknown quantities.
  3. Substitute, construct, or prove one justified step at a time.
  4. 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

Side-Splitter and Proportional Segments

Find, construct, prove, or explain each result. Show reasoning and label units.

U05 · L05
Name
Date
1

A dilation has scale factor 2. A preimage side is 6. Find the image side.

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: 12
Reasoning: Multiply by the scale factor: 6·2.
2

Triangles are similar. A side 5 corresponds to 8; a side 6 corresponds to x. Find x.

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: 9.6
Reasoning: Use 5/8=6/x.
3

Which angle evidence alone is sufficient for triangle similarity?

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: Two pairs of congruent angles (AA)
Reasoning: The third angle is then congruent automatically.
4

How does a dilation with scale factor 1/2 affect angle measure?

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: It preserves angle measure
Reasoning: Dilations change lengths proportionally but preserve angles.
5

A 1:50 drawing shows a wall 8 cm long. What is the actual length?

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: 400 cm or 4 m
Reasoning: Multiply the drawing length by 50.
6

Why should scale units be written explicitly?

Similarity geometry diagram A clear labeled diagram supporting similarity reasoning. smalldilated image
Answer: To prevent comparing unlike units and to make conversions auditable
Reasoning: A scale ratio is meaningful only when units are controlled.