Question
Dilations Practice
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Question
Use the image to answer the question.
A coordinate grid of quadrant 1 shows 2 plotted triangles of the same shape but different sizes. The x and y axes range from 0 to 10 in unit increments. A small tilted triangle upper A upper B upper C is formed by connecting the following plots: point upper A is left parenthesis 2 comma 1 right parenthesis; point upper B is left parenthesis 3 comma 2 right parenthesis; and point upper C is left parenthesis 1 comma 3 right parenthesis. A large tilted dotted triangle is formed by connecting the following plots: point upper A prime is left parenthesis 6 comma 3 right parenthesis; point upper B prime is left parenthesis 9 comma 6 right parenthesis; point upper C prime is left parenthesis 3 comma 9 right parenthesis.
What is the scale factor of a single dilation between the preimage ABC
and image A′B′C′
?
(1 point)
Complete this assessment to review what you've learned. It will not count toward your grade.
3 of 53 of 5 Items
Question
Use the image to answer the question.
A coordinate grid of quadrant 1 shows 2 plotted triangles of the same shape but different sizes. The x and y axes range from 0 to 10 in unit increments. A small tilted triangle upper A upper B upper C is formed by connecting the following plots: point upper A is left parenthesis 2 comma 1 right parenthesis; point upper B is left parenthesis 3 comma 2 right parenthesis; and point upper C is left parenthesis 1 comma 3 right parenthesis. A large tilted dotted triangle is formed by connecting the following plots: point upper A prime is left parenthesis 6 comma 3 right parenthesis; point upper B prime is left parenthesis 9 comma 6 right parenthesis; point upper C prime is left parenthesis 3 comma 9 right parenthesis.
What is the scale factor of a single dilation between the preimage ABC
and image A′B′C′
?
(1 point)
Answers
Answered by
GPT-4o mini
To find the scale factor of the dilation between the preimage triangle \( ABC \) and the image triangle \( A'B'C' \), we need to compare the lengths of corresponding sides of the two triangles.
First, we can calculate the lengths of one side of each triangle using the distance formula. Let's calculate \( AB \) for triangle \( ABC \) and \( A'B' \) for triangle \( A'B'C' \).
1. **Length of side \( AB \)**:
- Points \( A(2, 1) \) and \( B(3, 2) \)
\[
AB = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{2}
\]
2. **Length of side \( A'B' \)**:
- Points \( A'(6, 3) \) and \( B'(9, 6) \)
\[
A'B' = \sqrt{(9 - 6)^2 + (6 - 3)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
Now we can find the scale factor by taking the ratio of the lengths of the corresponding sides:
\[
\text{Scale factor} = \frac{\text{Length of } A'B'}{\text{Length of } AB} = \frac{3\sqrt{2}}{\sqrt{2}} = 3
\]
Thus, the scale factor of the dilation between \( ABC \) and \( A'B'C' \) is **3**.
First, we can calculate the lengths of one side of each triangle using the distance formula. Let's calculate \( AB \) for triangle \( ABC \) and \( A'B' \) for triangle \( A'B'C' \).
1. **Length of side \( AB \)**:
- Points \( A(2, 1) \) and \( B(3, 2) \)
\[
AB = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{2}
\]
2. **Length of side \( A'B' \)**:
- Points \( A'(6, 3) \) and \( B'(9, 6) \)
\[
A'B' = \sqrt{(9 - 6)^2 + (6 - 3)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
Now we can find the scale factor by taking the ratio of the lengths of the corresponding sides:
\[
\text{Scale factor} = \frac{\text{Length of } A'B'}{\text{Length of } AB} = \frac{3\sqrt{2}}{\sqrt{2}} = 3
\]
Thus, the scale factor of the dilation between \( ABC \) and \( A'B'C' \) is **3**.