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Math  /  Geometry

QuestionLesson 3 Practice Problems (1.) Pentagon A′B′C′D′E′A^{\prime} B^{\prime} C^{\prime} D^{\prime} E^{\prime} is the image of pentagon ABCDEA B C D E after a dilation centered at FF. What is the scale factor of this dilation?

Studdy Solution

STEP 1

1. Pentagon A′B′C′D′E′A'B'C'D'E' is a dilated image of pentagon ABCDEABCDE.
2. The dilation is centered at point FF.
3. The distances from FF to the vertices of ABCDEABCDE are either 33 or 66.
4. The corresponding distances from FF to the vertices of A′B′C′D′E′A'B'C'D'E' are either 55 or 1010.

STEP 2

1. Understand the concept of dilation and scale factor.
2. Identify corresponding distances from the center of dilation.
3. Calculate the scale factor using the corresponding distances.

STEP 3

Understand that a dilation involves resizing a figure by a scale factor, with respect to a center point. The scale factor is the ratio of a distance from the center of dilation to a point on the image to the corresponding distance to a point on the original figure.

STEP 4

Identify corresponding distances from the center of dilation FF. For example, if FF to a vertex of ABCDEABCDE is 33, then the corresponding distance from FF to the same vertex of A′B′C′D′E′A'B'C'D'E' is 55. Similarly, if FF to a vertex of ABCDEABCDE is 66, then the corresponding distance from FF to the same vertex of A′B′C′D′E′A'B'C'D'E' is 1010.

STEP 5

Calculate the scale factor using the ratio of corresponding distances. Use either pair of corresponding distances:
Scale Factor=Distance from F to a vertex of A′B′C′D′E′Distance from F to the corresponding vertex of ABCDE\text{Scale Factor} = \frac{\text{Distance from } F \text{ to a vertex of } A'B'C'D'E'}{\text{Distance from } F \text{ to the corresponding vertex of } ABCDE}
Using the distances 55 and 33:
Scale Factor=53\text{Scale Factor} = \frac{5}{3}
Or using the distances 1010 and 66:
Scale Factor=106=53\text{Scale Factor} = \frac{10}{6} = \frac{5}{3}
The scale factor of the dilation is:
53 \boxed{\frac{5}{3}}

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