2.1.1: Projecting and Scaling

Lesson

Let's explore scaling.

Exercise \(\PageIndex{1}\): Number Talk: Remembering Fraction Division

Find each quotient. Write your answer as a fraction or a mixed number.

\(6\frac{1}{4}\div 2\)

\(10\frac{1}{7}\div 5\)

\(8\frac{1}{2}\div 11\)

Exercise \(\PageIndex{2}\): Sorting Rectangles

Rectangles were made by cutting an \(8\frac{1}{2}\)-inch by 11-inch piece of paper in half, in half again, and so on, as illustrated in the diagram. Find the lengths of each rectangle and enter them in the appropriate table.

Figure \(\PageIndex{1}\): An image of an 8 point 5 by 11 inch rectangle. A dashed line divides the entire rectangle in half and one rectangle is labeled B. A dashed line divides the other half of B and one rectangle is labeled C. A dashed line divides the other half of C and one rectangle is labeled D. A dashed line divides the other half of D and both rectangles are labeled E.

1. Some of the rectangles are scaled copies of the full sheet of paper (Rectangle A). Record the measurements of those rectangles in this table.

Table \(\PageIndex{1}\)

rectangle	length of short side (inches)	length of long side (inches)
\(A\)	\(8\frac{1}{2}\)	\(11\)

2. Some of the rectangles are not scaled copies of the full sheet of paper. Record the measurements of those rectangles in this table.

Table \(\PageIndex{2}\)

rectangle	length of short side (inches)	length of long side (inches)

3. Look at the measurements for the rectangles that are scaled copies of the full sheet of paper. What do you notice about the measurements of these rectangles? Look at the measurements for the rectangles that are not scaled copies of the full sheet. What do you notice about these measurements?

4. Stack the rectangles that are scaled copies of the full sheet so that they all line up at a corner, as shown in the diagram. Do the same with the other set of rectangles. On each stack, draw a line from the bottom left corner to the top right corner of the biggest rectangle. What do you notice?

Figure \(\PageIndex{2}\)

5. Stack all of the rectangles from largest to smallest so that they all line up at a corner. Compare the lines that you drew. Can you tell, from the drawn lines, which set each rectangle came from?

Are you ready for more?

In many countries, the standard paper size is not 8.5 inches by 11 inches (called “letter” size), but instead 210 millimeters by 297 millimeters (called “A4” size). Are these two rectangle sizes scaled copies of one another?

Exercise \(\PageIndex{3}\): Scaled Rectangles

Here is a picture of Rectangle R, which has been evenly divided into smaller rectangles. Two of the smaller rectangles are labeled B and C.

1. Is \(B\) a scaled copy of \(R\)? If so, what is the scale factor?
2. Is \(C\) a scaled copy of \(B\)? If so, what is the scale factor?
3. Is \(C\) a scaled copy of \(R\)? If so, what is the scale factor?

Figure \(\PageIndex{3}\)

Summary

Scaled copies of rectangles have an interesting property. Can you see what it is?

Here, the larger rectangle is a scaled copy of the smaller one (with a scale factor of \(\frac{3}{2}\)). Notice how the diagonal of the large rectangle contains the diagonal of the smaller rectangle. This is the case for any two scaled copies of a rectangle if we line them up as shown. If two rectangles are not scaled copies of one another, then the diagonals do not match up. In this unit, we will investigate how to make scaled copies of a figure.

Figure \(\PageIndex{4}\): A small rectangle inside a larger rectangle with same lower left point and a dashed diagonal line from that lower left point through both upper points. The small rectangle has length 4 and height 2. The large rectangle has length 6 and height 3.

Glossary Entries

Definition: Scale Factor

To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.

In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).

Figure \(\PageIndex{5}\)

Practice

Exercise \(\PageIndex{4}\)

Rectangle \(A\) measures 12 cm by 3 cm. Rectangle \(B\) is a scaled copy of Rectangle \(A\). Select all of the measurement pairs that could be the dimensions of Rectangle \(B\).

1. \(6\) cm by \(1.5\) cm
2. \(10\) cm by \(2\) cm
3. \(13\) cm by \(4\) cm
4. \(18\) cm by \(4.5\) cm
5. \(80\) cm by \(20\) cm

Exercise \(\PageIndex{5}\)

Rectangle \(A\) has length 12 and width 8. Rectangle \(B\) has length 15 and width 10. Rectangle \(C\) has length 30 and width 15.

1. Is Rectangle \(A\) a scaled copy of Rectangle \(B\)? If so, what is the scale factor?
2. Is Rectangle \(B\) a scaled copy of Rectangle \(A\)? If so, what is the scale factor?
3. Explain how you know that Rectangle \(C\) is not a scaled copy of Rectangle \(B\).
4. Is Rectangle \(A\) a scaled copy of Rectangle \(C\)? If so, what is the scale factor?

Exercise \(\PageIndex{6}\)

Here are three polygons.

Figure \(\PageIndex{6}\)

1. Draw a scaled copy of Polygon A with scale factor \(\frac{1}{2}\).
2. Draw a scaled copy of Polygon B with scale factor \(2\).
3. Draw a scaled copy of Polygon C with scale factor \(\frac{1}{4}\).

Exercise \(\PageIndex{7}\)

Which of these sets of angle measures could be the three angles in a triangle?

1. \(40^{\circ}, 50^{\circ}, 60^{\circ}\)
2. \(50^{\circ}, 60^{\circ}, 70^{\circ}\)
3. \(60^{\circ}, 70^{\circ}, 80^{\circ}\)
4. \(70^{\circ}, 80^{\circ}, 90^{\circ}\)

From (Unit 1.4.2)

Exercise \(\PageIndex{8}\)

In the picture lines \(AB\) and \(CD\) are parallel. Find the measures of the following angles. Explain your reasoning.

Figure \(\PageIndex{7}\): Three lines in a plane. Line A B. Line B F. Line D E. Lines B F and A B intersect at the point B. Lines B F and D E intersect at point C. Line D E is above line A B. Angle A B F is labeled 38 degrees.

1. \(\angle BCD\)
2. \(\angle ECF\)
3. \(\angle DCF\)

(From Unit 1.4.1)