Unit 1 Scale Drawings Comprehensive Assessment Review Study Notes

Properties of Scaled Copy Quadrilaterals and Angle Invariance

  • Definition of Scaled Copies: A figure is a scaled copy of another if all corresponding edge lengths are multiplied by the exact same scale factor, while all corresponding interior angle measures remain identical.
  • Corresponding Side Lengths and Scale Factor Calculation:
    • Consider Quadrilateral ABCDABCD and Quadrilateral EFGHEFGH, where ABCDABCD is a scaled copy of EFGHEFGH
    • Length of segment AB=3AB = 3
    • Length of segment BC=2BC = 2
    • Length of segment CD=4CD = 4
    • Length of segment DA=5DA = 5
    • Length of segment EF=6EF = 6
    • Length of segment FG=4FG = 4
    • Length of segment GH=8GH = 8
    • Length of segment HE=10HE = 10
    • The scale factor from EFGHEFGH to ABCDABCD is k=12=0.5k = \frac{1}{2} = 0.5
    • The scale factor from ABCDABCD to EFGHEFGH is k=2k = 2
  • Analysis of Geometric Statements:
    • Segment Relationship BCBC to EFEF: Segment BCBC has a length of 22 units and segment EFEF has a length of 66 units. Thus, BC=13EFBC = \frac{1}{3} EF, making the assertion that BCBC is half of EFEF false.
    • Segment Relationship CDCD to GHGH: Segment CDCD has a length of 44 units and segment GHGH has a length of 88 units. Therefore, CD=12GHCD = \frac{1}{2} GH, which is a true statement.
    • Angle Invariance: The measure of angle ADCADC is equal to the measure of angle EHGEHG. Angles of scaled copies are strictly preserved and do not change regardless of the scale factor applied.
    • Segment Length ABAB: The length of segment ABAB is confirmed to be 33 units, which is half of its corresponding segment EF=6EF = 6
    • Area Scaling Principle: The area of a scaled copy does not scale by the linear scale factor kk, but by k2k^2. Since the linear scale factor from EFGHEFGH to ABCDABCD is 12\frac{1}{2}, the area of ABCDABCD is (12)2=14\big(\frac{1}{2}\big)^2 = \frac{1}{4} of the area of EFGHEFGH, making the claim that its area is half false.

Identifying Scaled Rectangles and Scale Factors

  • Aspect Ratio Maintenance: For a rectangle to be a valid scaled copy of another, both the length and width must be multiplied by the same scale factor kk
  • Reference Dimensions: Rectangle AA measures 8 inches8\text{ inches} by 2 inches2\text{ inches} (aspect ratio of 4:14:1).
  • Evaluation of Scale Dimension Options:
    • 40 inches40\text{ inches} by 10 inches10\text{ inches}:     Scale factor length=408=5\text{Scale factor length} = \frac{40}{8} = 5Scale factor width=102=5\text{Scale factor width} = \frac{10}{2} = 5     Since both factors equal 55, this option is a valid scaled copy.
    • 10 inches10\text{ inches} by 2.5 inches2.5\text{ inches}:     Scale factor length=108=1.25\text{Scale factor length} = \frac{10}{8} = 1.25Scale factor width=2.52=1.25\text{Scale factor width} = \frac{2.5}{2} = 1.25     Since both factors equal 1.251.25, this option is a valid scaled copy.
    • 9 inches9\text{ inches} by 3 inches3\text{ inches}:     Scale factor length=98=1.125\text{Scale factor length} = \frac{9}{8} = 1.125Scale factor width=32=1.5\text{Scale factor width} = \frac{3}{2} = 1.5     Since the scale factors are unequal (1.125≠1.51.125 \neq 1.5), this option is not a scaled copy.
    • 7 inches7\text{ inches} by 1 inch1\text{ inch}:     Scale factor length=78=0.875\text{Scale factor length} = \frac{7}{8} = 0.875Scale factor width=12=0.5\text{Scale factor width} = \frac{1}{2} = 0.5     Since the scale factors are unequal (0.875≠0.50.875 \neq 0.5), this option is not a scaled copy.
    • 6.4 inches6.4\text{ inches} by 1.6 inches1.6\text{ inches}:     Scale factor length=6.48=0.8\text{Scale factor length} = \frac{6.4}{8} = 0.8Scale factor width=1.62=0.8\text{Scale factor width} = \frac{1.6}{2} = 0.8     Since both factors equal 0.80.8, this option is a valid scaled copy.

Calculating Actual Areas from Scale Drawings

  • Scale Drawing Specifications:
    • Scale drawing length: 6 inches6\text{ inches}
    • Scale drawing width: 5 inches5\text{ inches}
    • Actual rectangular parking lot length: 240 feet240\text{ feet}
  • Step-by-Step Area Calculation:
    • Step 1: Determine the linear scale multiplier for feet per inch.     Linear Scale=240 feet6 inches=40 feet/inch\text{Linear Scale} = \frac{240\text{ feet}}{6\text{ inches}} = 40\text{ feet/inch}
    • Step 2: Calculate the actual width of the parking lot using the derived scale multiplier.     Actual Width=5 inches×40 feet/inch=200 feet\text{Actual Width} = 5\text{ inches} \times 40\text{ feet/inch} = 200\text{ feet}
    • Step 3: Calculate the actual area of the parking lot.     Actual Area=Actual Length×Actual Width\text{Actual Area} = \text{Actual Length} \times \text{Actual Width}Actual Area=240 feet×200 feet=48,000 square feet\text{Actual Area} = 240\text{ feet} \times 200\text{ feet} = 48,000\text{ square feet}
  • Summary of Option Analysis:
    • Option A: 48,000 sq ft48,000\text{ sq ft} (Correct)
    • Option B: 1,200 sq ft1,200\text{ sq ft}
    • Option C: 200 sq ft200\text{ sq ft}
    • Option D: 30 sq ft30\text{ sq ft}

Polygon Scaling and Scale Factor Transformations

  • Linear Scaling Rules:
    • Scaling a polygon by a scale factor of k=3k = 3 requires multiplying every original side length by 33
    • Interior angles remain unchanged during scaling.
  • Edge Transformation Details:
    • An original segment of length 1 unit1\text{ unit} transforms to 1×3=3 units1 \times 3 = 3\text{ units}
    • An original segment of length 2 units2\text{ units} transforms to 2×3=6 units2 \times 3 = 6\text{ units}
    • An original segment of length 3 units3\text{ units} transforms to 3×3=9 units3 \times 3 = 9\text{ units}

Solving Map Scale Proportions

  • Base Map Scale:
    • Scale ratio: 8 cm8\text{ cm} on the map represents 5 km5\text{ km} of actual distance (8 cm:5 km8\text{ cm} : 5\text{ km}).
  • Map Distance Calculation for Actual Distance:
    • Target actual distance: 8 km8\text{ km}
    • Proportional equation:     8 cm5 km=x cm8 km\frac{8\text{ cm}}{5\text{ km}} = \frac{x\text{ cm}}{8\text{ km}}
    • Cross-multiplication and solution:     5x=8×85x = 8 \times 85x=645x = 64x=645=12.8 cmx = \frac{64}{5} = 12.8\text{ cm}
    • An actual distance of 8 km8\text{ km} is represented by 12.8 cm12.8\text{ cm} on the map.
  • Actual Distance Calculation for Map Distance:
    • Target map distance: 5 cm5\text{ cm}
    • Proportional equation:     8 cm5 km=5 cmx km\frac{8\text{ cm}}{5\text{ km}} = \frac{5\text{ cm}}{x\text{ km}}
    • Cross-multiplication and solution:     8x=5×58x = 5 \times 58x=258x = 25x=258=3.125 kmx = \frac{25}{8} = 3.125\text{ km}
    • A map distance of 5 cm5\text{ cm} represents an actual distance of 3.125 km3.125\text{ km}.

Multi-Map Scale Conversions

  • First Map Parameters:
    • Map 1 Scale: 1 cm1\text{ cm} to 20 km20\text{ km} (1 cm:20 km1\text{ cm} : 20\text{ km})
    • Distance between Fresno and San Francisco on Map 1: 15 cm15\text{ cm}
  • Actual Ground Distance Calculation:   Actual Distance=15 cm×20 km/cm=300 km\text{Actual Distance} = 15\text{ cm} \times 20\text{ km/cm} = 300\text{ km}
  • Second Map Parameters and Conversion:
    • Map 2 Scale: 1 cm1\text{ cm} to 100 km100\text{ km} (1 cm:100 km1\text{ cm} : 100\text{ km})
    • Distance on Map 2:     Distance on Map 2=300 km100 km/cm=3 cm\text{Distance on Map 2} = \frac{300\text{ km}}{100\text{ km/cm}} = 3\text{ cm}
  • Reasoning Process:
    • The actual distance between Fresno and San Francisco is first determined using Map 1 as 15 cm×20 km/cm=300 km15\text{ cm} \times 20\text{ km/cm} = 300\text{ km}.
    • That actual distance is then converted into the scale of Map 2 by dividing 300 km300\text{ km} by 100 km/cm100\text{ km/cm}, yielding 3 cm3\text{ cm}.

Architectural Scale Drawings and Unit Conversions

  • Kitchen Floor Plan Parameters:
    • Actual kitchen dimensions: Length = 6 m6\text{ m}, Width = 2 m2\text{ m}
    • Scale ratio chosen: 1:401 : 40
    • Equipment required: Centimeter ruler
  • Dimension Calculations for Scale Drawing (1:40):
    • Unit Conversion to Centimeters:     Actual Length=6 m=600 cm\text{Actual Length} = 6\text{ m} = 600\text{ cm}Actual Width=2 m=200 cm\text{Actual Width} = 2\text{ m} = 200\text{ cm}
    • Scale Drawing Length:     Drawing Length=600 cm40=15 cm\text{Drawing Length} = \frac{600\text{ cm}}{40} = 15\text{ cm}
    • Scale Drawing Width:     Drawing Width=200 cm40=5 cm\text{Drawing Width} = \frac{200\text{ cm}}{40} = 5\text{ cm}
    • The scale drawing of the kitchen floor plan must be drawn with dimensions 15 cm15\text{ cm} by 5 cm5\text{ cm}.
  • Door Width Scaling:
    • Actual kitchen door width: 1.2 m=120 cm1.2\text{ m} = 120\text{ cm}
    • Scale drawing door width calculation:     Drawing Door Width=120 cm40=3 cm\text{Drawing Door Width} = \frac{120\text{ cm}}{40} = 3\text{ cm}
    • Explanation: Because the chosen scale is 1:401 : 40, every feature on the scale drawing is 140\frac{1}{40} of its actual size. Dividing 120 cm120\text{ cm} by 4040 gives 3 cm3\text{ cm}.
  • Kitchen Table Actual Measurements Calculation:
    • Table scale drawing measurements: 4 cm4\text{ cm} by 2.5 cm2.5\text{ cm}
    • Actual Length Calculation:     Actual Table Length=4 cm×40=160 cm=1.6 m\text{Actual Table Length} = 4\text{ cm} \times 40 = 160\text{ cm} = 1.6\text{ m}
    • Actual Width Calculation:     Actual Table Width=2.5 cm×40=100 cm=1 m\text{Actual Table Width} = 2.5\text{ cm} \times 40 = 100\text{ cm} = 1\text{ m}
    • The actual measurements of the kitchen table are 1.6 m1.6\text{ m} by 1 m1\text{ m} (or 160 cm160\text{ cm} by 100 cm100\text{ cm}).