Scale Drawings - Geometry in Two Dimensions (Pre-Algebra)

Denoting a Scale
Ratio	$1 in : 100 ft$
Equals Sign	$1 in = 100 ft$
Fraction	$\frac{1 in}{1 0 0 ft}$

Denoting a Scale

Ratio

1 in : 100 ft

Equals Sign

1 in = 100 ft

Fraction

\frac{1 in}{1 0 0 ft}

Place $1$	Place $2$	Distance on the Map (in)	Actual Distance (mi)
Bakery	Bank	$2.85$	$0.75$
Bank	Aunt's house	$x$	$0.70$

Place

1

Place

2

Distance on the Map (in)

Actual Distance (mi)

Bakery

Bank

2.85

0.75

Bank

Aunt's house

x

0.70

	Scale Model	Lighthouse
Width	$10 cm$	$5.25 m$
Height	$30 cm$	$?$

Scale Model

Lighthouse

Width

10 cm

5.25 m

Height

30 cm

?

The length scale factor of the blueprint is the ratio of a dimension on the blueprint to the corresponding actual dimension. The dimensions of the bed on the blueprint are written on its side. Additionally, Kevin measured his actual bed length.

	Blueprint Dimensions (cm)	Actual Dimensions (m)
Bed length	$9$	$1.80$
Bed width	$5$

Before finding the length scale factor, all the dimensions should be written with the same units of measure. Use the fact that

1

meter is the same as

100

centimeters to convert

1.80

meters to centimeters.

1.80 m \cdot \frac{1 0 0 cm}{1 m} = 180 cm

Next, divide the bed length on the blueprint by the actual length of the bed to find the scale factor.

Scale factor 1 = \frac{9}{1 8 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 9}{b / 9}$

Scale factor 1 = \frac{1}{2 0}

The length scale factor of the blueprint is

\frac{1}{2 0} .

This means that the actual dimensions of the bed — and the entire room — were divided by

20

to make the blueprint.

Similarly, the length scale factor of the scale shed can be found. The width and length of the model are known and Kevin measured the actual width of the shed.

	Scale Model Dimensions (cm)	Actual Dimensions (m)
Length	$20$
Width	$15$	$4.50$

As before, write all the dimensions using the same units of measure.

4.50 m \cdot \frac{1 0 0 cm}{1 m} = 450 cm

The length scale factor is the quotient between the width of the scale shed and the actual width of the shed.

Scale factor 2 = \frac{1 5}{4 5 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 1 5}{b / 1 5}$

Scale factor 2 = \frac{1}{3 0}

Now that both length scale factors are known, the pin of the padlock can be found. Recall, it is

156

times the sum of the length scale factors.

Pin = 156 (scale factor 1 + scale factor 2)

SubstituteValues

Substitute values

Pin = 156 (\frac{1}{2 0} + \frac{1}{3 0})

▼

Simplify

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3 0}{b \cdot 3 0}$

Pin = 156 (\frac{1 \cdot 3 0}{2 0 \cdot 3 0} + \frac{1}{3 0})

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2 0}{b \cdot 2 0}$

Pin = 156 (\frac{1 \cdot 3 0}{2 0 \cdot 3 0} + \frac{1 \cdot 2 0}{3 0 \cdot 2 0})

Multiply

Pin = 156 (\frac{3 0}{6 0 0} + \frac{2 0}{6 0 0})

AddFrac

Add fractions

Pin = 156 \cdot \frac{5 0}{6 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 5 0}{b / 5 0}$

Pin = 156 \cdot \frac{1}{1 2}

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

Pin = \frac{1 5 6}{1 2}

CalcQuot

Calculate quotient

Pin = 13

The pin to open the padlock is

13 .

Area of Original Triangle	Length Scale Factor	Area of Scale Triangle	Ratio of Areas
$2.8$	$2$	$11.2$	$\frac{1 1 . 2}{2 . 8} = 4 = 2^{2}$
$2.8$	$0.5 = \frac{1}{2}$	$0.7$	$\frac{0 . 7}{2 . 8} = 0.25 = \frac{1}{4} = (\frac{1}{2})^{2}$
$2.8$	$1.5 = \frac{3}{2}$	$6.3$	$\frac{6 . 3}{2 . 8} = 2.25 = \frac{9}{4} = (\frac{3}{2})^{2}$

Area of Original Triangle

Length Scale Factor

Area of Scale Triangle

Ratio of Areas

2.8

2

11.2

\frac{1 1 . 2}{2 . 8} = 4 = 2^{2}

2.8

0.5 = \frac{1}{2}

0.7

\frac{0 . 7}{2 . 8} = 0.25 = \frac{1}{4} = (\frac{1}{2})^{2}

2.8

1.5 = \frac{3}{2}

6.3

\frac{6 . 3}{2 . 8} = 2.25 = \frac{9}{4} = (\frac{3}{2})^{2}

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Catch-Up and Review

Maps and Actual Distances

The Scale of a Map

Scale

Scale Drawing

Finding a Missing Length

Hint

Solution

Scale for Three Dimensional Objects

Scale Model

Height of a Lighthouse

Hint

Solution

Length Scale Factor

Finding Length Scale Factors

Hint

Solution

Next Stop

Hint

Solution

Length in a Scale Model

Hint

Solution

Relating the Area of a Scale Drawing to the Actual Area

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