Comprehensive Study Guide on Scale Drawings and Scaled Copies

Principles of Scale Drawings and Scaled Copies

• Definition of Scaled Copies:

  • A figure is considered a scaled copy of another figure if every corresponding length in the copy is produced by multiplying the original length by a single positive number, called the scale factor ($k$).
  • All corresponding internal angles in scaled copies remain equal (congruent).
  • The ratio between corresponding side lengths across scaled figures remains constant.

• Identifying Scaled Copies of Rectangles (Question 1):

  • Original Rectangle A Dimensions:
  • Width = $2\,\text{m}$
  • Length = $8\,\text{m}$
  • Side Length Ratio = $\frac{2\,\text{m}}{8\,\text{m}} = \frac{1}{4}$
  • Evaluation of Candidate Rectangles:
  • Rectangle B ($3\,\text{m} \times 12\,\text{m}$):
    • Ratio = $\frac{3\,\text{m}}{12\,\text{m}} = \frac{1}{4}$
    • Scale factor from A = $1.5$ ($2 \times 1.5 = 3$ and $8 \times 1.5 = 12$)
    • Status: Scaled Copy
  • Rectangle C ($4\,\text{m} \times 16\,\text{m}$):
    • Ratio = $\frac{4\,\text{m}}{16\,\text{m}} = \frac{1}{4}$
    • Scale factor from A = $2$ ($2 \times 2 = 4$ and $8 \times 2 = 16$)
    • Status: Scaled Copy
  • Rectangle D ($1\,\text{m} \times 4\,\text{m}$):
    • Ratio = $\frac{1\,\text{m}}{4\,\text{m}} = \frac{1}{4}$
    • Scale factor from A = $0.5$ ($2 \times 0.5 = 1$ and $8 \times 0.5 = 4$)
    • Status: Scaled Copy
  • Rectangle E ($3\,\text{m} \times 9\,\text{m}$):
    • Ratio = $\frac{3\,\text{m}}{9\,\text{m}} = \frac{1}{3} \neq \frac{1}{4}$
    • Status: Not a Scaled Copy
  • Rectangle F ($7\,\text{m} \times 11\,\text{m}$):
    • Ratio = $\frac{7\,\text{m}}{11\,\text{m}} \neq \frac{1}{4}$
    • Status: Not a Scaled Copy
  • Conclusion: Rectangles B, C, and D are scaled copies of Rectangle A.

Properties of Scaled Geometric Figures

• Figure Similarity Relationships (Question 2):

  • Given: Figure $EFGH$ is a scaled copy of figure $ABCD$.
  • Analysis of Statements:
  • Statement A: Segment GH is three times as long as segment AB.
    • Analysis: Incorrect; segment $GH$ corresponds to segment $CD$, not $AB$.
  • Statement B: The ratio of $\frac{AB}{BC}$ is equivalent to the ratio of $\frac{EF}{FG}$.
    • Analysis: True; internal ratio of corresponding sides is always preserved in scaled copies.
  • Statement C: The scale factor from EFGH to ABCD is $\frac{1}{3}$.
    • Analysis: True; if figure $EFGH$ is enlarged from $ABCD$ by a scale factor of $3$, going from $EFGH$ back to $ABCD$ requires scaling by the reciprocal, $\frac{1}{3}$.
  • Statement D: The length of segment BC is 2 units.
    • Analysis: Evaluated based on corresponding side proportions provided in the geometric diagram.
  • Statement E: The area of EFGH is three times the area of ABCD.
    • Analysis: False; when side lengths scale by a factor of $k = 3$, the area scales by $k^2 = 3^2 = 9$ times the original area.

• Constructing Scaled Copies with Scale Factor 1 (Question 4):

  • Definition: A scale factor of $k = 1$ produces a congruent figure identical in size and shape to the original polygon.
  • Polygon Side Dimensions:
  • Top horizontal segment = $12$
  • Left vertical segment = $8$
  • Top right segment = $12$
  • Right vertical segment = $20$
  • Bottom horizontal segment = $20$
  • Scaled Copy Dimensions ($k = 1$):
  • Top side = $12 \times 1 = 12$
  • Left side = $8 \times 1 = 8$
  • Top right segment = $12 \times 1 = 12$
  • Right side = $20 \times 1 = 20$
  • Bottom side = $20 \times 1 = 20$

Map Scale Calculations and Proportion Analysis

• Town Map Distance Problems (Question 5):

  • Given Scale Ratio: $2\,\text{cm}$ on the map represents $5\,\text{km}$ in actual distance ($2\,\text{cm} = 5\,\text{km}$).
  • Part A: Determining Map Distance from Actual Distance:
  • Scenario: School and hospital are $15\,\text{km}$ apart in actual distance.
  • Step 1: Divide total actual distance by the scale distance unit: $\frac{15\,\text{km}}{5\,\text{km}} = 3$ scale units.
  • Step 2: Multiply scale units by map representation per unit: $3 \times 2\,\text{cm} = 6\,\text{cm}$.
  • Result: The distance between the school and the hospital on the map is $6\,\text{cm}$.
  • Part B: Determining Actual Distance from Map Distance:
  • Scenario: School and playground are $8\,\text{cm}$ apart on the map.
  • Step 1: Divide total map distance by scale map unit: $\frac{8\,\text{cm}}{2\,\text{cm}} = 4$ scale units.
  • Step 2: Multiply scale units by actual distance per unit: $4 \times 5\,\text{km} = 20\,\text{km}$.
  • Result: The actual distance between the school and the playground is $20\,\text{km}$.

• Comparison Table of Map to Actual Distances:

  • Map Distance $2\,\text{cm}$ $\rightarrow$ Actual Distance $5\,\text{km}$
  • Map Distance $6\,\text{cm}$ $\rightarrow$ Actual Distance $15\,\text{km}$
  • Map Distance $8\,\text{cm}$ $\rightarrow$ Actual Distance $20\,\text{km}$

Map Scale Comparison and Size Scaling

• Trail Map Comparison Mechanics (Question 6):
  • Old Map Scale: $1\,\text{cm} = 400\,\text{m}$
  • New Map Scale: $1\,\text{cm} = 100\,\text{m}$
  • Part A: Size Comparison of the Maps:
  • Question: If the maps represent the same area, will the new map be larger, smaller, or the same size as the old map?
  • Selected Option: A. Larger
  • Explanation: Because each centimeter on the new map represents a smaller ground distance ($100\,\text{m}$) than on the old map ($400\,\text{m}$), more centimeters are needed on the new map to show the same actual distance ($400 / 100 = 4$ times as many centimeters). Consequently, the overall drawing of the trail and the surrounding area on the new map will be four times as long in each dimension, making the new map larger.
  • Part B: Trail Length Calculation on New Map:
  • Given: Trail length on old map = $20\,\text{cm}$.
  • Method 1 (Actual Trail Distance):
    • Calculate actual length of the trail: $\text{Actual Length} = 20\,\text{cm} \times 400\,\text{m/cm} = 8,000\,\text{m}$.
    • Calculate map length on new map: $\text{New Map Length} = \frac{8,000\,\text{m}}{100\,\text{m/cm}} = 80\,\text{cm}$.
  • Method 2 (Direct Scale Ratio):
    • Scale factor between maps: $\frac{\text{Old Scale}}{\text{New Scale}} = \frac{400\,\text{m/cm}}{100\,\text{m/cm}} = 4$
    • Calculate new map length: $20\,\text{cm} \times 4 = 80\,\text{cm}$.
  • Result: The running trail is $80\,\text{cm}$ long on the new map.

Scale Drawing Area Calculations

• Rectangular Park Area Analysis (Question 3):
  • Given Scale Drawing Dimensions:
  • Drawing Width = $8\,\text{in}$
  • Drawing Length = $11\,\text{in}$
  • Actual Park Length = $330\,\text{yd}$
  • Step-by-Step Area Calculation:
  • Step 1: Determine the Linear Scale Factor:
    • $\text{Linear Scale} = \frac{330\,\text{yd}}{11\,\text{in}} = 30\,\text{yd/in}$
    • Each inch on the scale drawing represents $30\,\text{yards}$ in reality.
  • Step 2: Calculate Actual Park Width:
    • $\text{Actual Width} = 8\,\text{in} \times 30\,\text{yd/in} = 240\,\text{yd}$
  • Step 3: Calculate Actual Area of the Park:
    • $\text{Actual Area} = \text{Actual Width} \times \text{Actual Length}$
    • $\text{Actual Area} = 240\,\text{yd} \times 330\,\text{yd} = 79,200\,\text{square yards}$
  • Multiple-Choice Evaluation:
  • A. $88\,\text{square yards}$ (Incorrect: Represents $\text{drawing width} \times \text{drawing length} = 8 \times 11$
  • B. $240\,\text{square yards}$ (Incorrect: Represents actual width in yards)
  • C. $2,640\,\text{square yards}$ (Incorrect: Represents $8 \times 330$)
  • D. $79,200\,\text{square yards}$ (Correct Answer)

Curriculum Context and Specifications

• Curriculum Information:
  • Program: Amplify Desmos Math
  • Instructional Foundations: Illustrative Mathematics (IM) and Open Up Resources
  • Module/Unit Focus: Unit 1 Study Guide - Scale Drawings (Unit 7.1, Form B)