Question

for exercises 1 - 4, determine the dimensions of the object after the given dilation.

1. a square with side length of 2 feet dilated by a scale factor of 3
2. a rectangle with sides measuring 4 inches and 10 inches dilated by a scale factor of \\( \frac { 1 } { 2 } \\)
3. a kite dilated by a scale factor of 3
4. a triangle with sides measuring 6 centimeters, 27 centimeters, and 24 centimeters dilated by a scale factor of \\( \frac { 2 } { 3 } \\)

for exercises 5 - 7, determine the scale factor used in each dilation.

1. a square with side length 5 centimeters is dilated to create a square with sidelength 45 centimeters
2. a triangle with sides measuring 3, 8, and 6 inches is dilated to create a triangle with sides measuring 10.5, 28, and 21 inches
3. a quadrilateral with sides measuring 12, 4, 6, and 8 meters is dilated to create a quadrilateral with sides measuring 9, 3, \\( \frac { 9 } { 2 } \\), and 6 meters

for exercises 8 and 9, answer the questions.

1. what is another word that can be used in the place of the word dilation?

a. spin
b. flip
c. stretch
d. slide

1. how would you explain to a classmate what it means when the scale factor of a dilation is less than 1? equal to 1? greater than 1?

Explanation:

Step1: Solve problem 1

The side length of the square is \(2\) feet. After dilation by a scale factor of \(3\), the new side length is \(2\times3 = 6\) feet.

Step2: Solve problem 2

For the rectangle with sides \(4\) inches and \(10\) inches, after dilation by a scale factor of \(\frac{1}{2}\), the new side lengths are \(4\times\frac{1}{2}=2\) inches and \(10\times\frac{1}{2} = 5\) inches.

Step3: Solve problem 3

For the kite with side lengths \(7.8\) and \(9.3\), after dilation by a scale factor of \(3\), the new side lengths are \(7.8\times3=23.4\) and \(9.3\times3 = 27.9\).

Step4: Solve problem 4

For the triangle with sides \(6\) cm, \(27\) cm, and \(24\) cm, after dilation by a scale factor of \(\frac{2}{3}\), the new side lengths are \(6\times\frac{2}{3}=4\) cm, \(27\times\frac{2}{3}=18\) cm, and \(24\times\frac{2}{3}=16\) cm.

Step5: Solve problem 5

The scale factor \(k=\frac{45}{5}=9\).

Step6: Solve problem 6

Take one pair of corresponding sides, say \(3\) and \(10.5\). The scale factor \(k=\frac{10.5}{3}=3.5\).

Step7: Solve problem 7

Take one pair of corresponding sides, say \(12\) and \(9\). The scale factor \(k=\frac{9}{12}=\frac{3}{4}\).

Step8: Solve problem 8

Dilation is a transformation that changes the size of a figure. Among the options, “stretch” (option c) is a more appropriate word as it implies a change in size. “Spin” (a) is rotation, “flip” (b) is reflection, and “slide” (d) is translation.

Step9: Solve problem 9

• If the scale factor \(k<1\), the dilated figure is smaller than the original figure (contraction).
• If \(k = 1\), the dilated figure is congruent to the original figure (no change in size).
• If \(k>1\), the dilated figure is larger than the original figure (enlargement).

Answer:

1. Side length of the square: \(6\) feet.
2. Side lengths of the rectangle: \(2\) inches and \(5\) inches.
3. Side lengths of the kite: \(23.4\) and \(27.9\).
4. Side lengths of the triangle: \(4\) cm, \(18\) cm, and \(16\) cm.
5. Scale factor: \(9\).
6. Scale factor: \(3.5\).
7. Scale factor: \(\frac{3}{4}\).
8. C. stretch.
9. If \(k<1\), the figure is smaller (contraction); if \(k = 1\), the figure is congruent (no size - change); if \(k>1\), the figure is larger (enlargement).