Study guide: Distances and Scaled Designs
Use right triangles, coordinates, and similarity to reason about fictional designs.
Ideas to explain in your own words
- Use a right-triangle relationship and check which side is the hypotenuse.
- Build a right triangle from coordinate differences.
- Use one scale factor for corresponding sides of similar figures.
- Explain why perimeter scales by a factor and area by its square.
Watch your reasoning
- 6² + 8² = 36 + 64 = 100, so the diagonal is √100 = 10 units. The squared quantity 100 is not itself the length.
- The horizontal difference is 8 and the vertical difference is 6. d = √(8² + 6²) = √100 = 10 units.
- The factor is 15 ÷ 6 = 2.5. Multiplying the corresponding 8-unit leg gives 8 × 2.5 = 20 units. The hypotenuse would be 25.
- The new sides are 8 and 20, so area is 8 × 20 = 160 square units. This is 4² = 16 times the original area of 10.
A worked review example
Find a diagonal with the Pythagorean theorem: 3² + 4² = 9 + 16 = 25, so c = √25 = 5 units. The diagonal is longer than either side but shorter than the 3 + 4 = 7 unit path along the edges. Find a Distance Between Coordinates: d = √[(4 + 2)² + (9 − 1)²] = √(36 + 64) = 10 units. Reversing the differences gives (−6)² + (−8)², the same squared distance. Scale Corresponding Lengths: The enlarged triangle has sides 10, 24, 26. Ratios 10/5, 24/12, and 26/13 all equal 2. Also 100 + 576 = 676 = 26², so the right-angle relationship is preserved. Separate Length Scaling from Area Scaling: Original area is 4 × 7 = 28 square units. New area is 12 × 21 = 252 = 9 × 28. New perimeter is 2(12 + 21) = 66 = 3 × 22 units.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.