Math 7 ( Period 4)

Scale Drawings & Models  7.3

A scale drawing is a diagram of an object in which the length and the width are proportional to the actual length and width of the object

The scale for the drawing gives the relationship between the drawing’s measurement and the actual measurements, such as “ 1 in : 2ft.” 

This scale means that 1 inch on the drawing represents the actual distance of 2 feet.

The scale factor for a scale drawing is the ratio of the length and width in the drawing to the corresponding actual length and width (in the same units of measurements)  for instance a sale of “.25 in : 2 ft” has a scale factor of  1/96 because you need to change either the inches to feet in the numerator  or the feet to inches in the denominator to have the same units of measurement. 

2 ft is equivalent to 24 inches so you have  .25/24 when you simplify that you get 1/96

Scale ratios are usually rates… They have two different labels.
1.5 in : 1000 miles.

Scale rates have NO LABELS at all because both the model and the actual are expressed in the same label.  The scale rate no longer has any labels and simplifies.

Using cross products you find that the missing  side is 12 inches

ALWAYS SET UP THE PROPORTION WITH 2 KNOWN MEASUREMENTS.

We looked at the scale drawing of the theater stage on Page 328 and found the length of the back wall. 
In the scale drawing .25 inches  = 2 ft  . 

The back wall in the drawing measures 1.5 inches  Let x = the actual length of the back wall

We set up a proportion

 

You could simplify BEFORE you cross multiply
but without you get  .25x = 2(1.5)

divide both sides by .25

x = 12
12 feet

What if you had changed the scale to a scale  with the same units of measures first? That is changed the “.25 inches = 2 feet” to “.25 inches = 24 inches”   So

Notice the difference – now everything is in inches and when you solve it you get    x = 144

BUT THAT IS IN INCHES… 

and 144 inches = 12 feet… the same answer you arrived at—without having to change to the same units of measure!!

In Geometry the definition of similar polygons is that they have proportional sides and congruent (equal) angles. The congruent angels keep the shape the same. The proportional sides make the polygons smaller or larger. 

So if you know that 2 triangles are similar, you can find a missing measurement of a side by setting up a proportion.

Example: You know that 2 triangles are similar. One is larger than the other You know the measurement of 2 sides of the smaller  are 3 inches and 4 inches. You know that the larger triangles corresponding shorter side is 9 inches.

COMPARE PERIMETER AND AREA

Perimeter of a rectangle is 2l + 2w

So if a photo is 8 inches by 10 inches, the perimeter is
2(8) + 2(10) = 36 inches

If you reduce the size of the picture 50% on the copier
P = 2(4) + 2(5) = 18. So the perimeter is 50% smaller

Area of a rectangle = lw = 8(10) = 80 inches squared
Is the reduced photo’s area 40 inches squared?

NO 4(5) = 20 inches squares.  WHY?

each of the sides is ½ the original so (1/2)(1/2) = ¼ of the original area..

 20 is 1/4 of the original area of 80