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The endpoints of [tex]$\overline{AB}$[/tex] are [tex]$A(2,2)$[/tex] and [tex]$B(3,8)$[/tex]. [tex]$\overline{AB}$[/tex] is dilated by a scale factor of 3.5 with the origin as the center of dilation to give the image [tex]$\overline{A^{\prime}B^{\prime}}$[/tex]. What are the slope [tex]$(m)$[/tex] and length of [tex]$\overline{A^{\prime}B^{\prime}}$[/tex]? Use the distance formula to help you decide: [tex]$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$[/tex].

A. [tex]$m=21, A^{\prime}B^{\prime}=\sqrt{37}$[/tex]

B. [tex]$m=6, A^{\prime}B^{\prime}=3.5\sqrt{37}$[/tex]

C. [tex]$m=6, A^{\prime}B^{\prime}=\sqrt{37}$[/tex]

D. [tex]$m=21, A^{\prime}B^{\prime}=3.5\sqrt{37}$[/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to find the slope [tex]\( m \)[/tex] and the length of the line segment [tex]\( \overline{A'B'} \)[/tex], which is the dilated image of [tex]\( \overline{AB} \)[/tex].

### Step 1: Calculate the Slope [tex]\( m \)[/tex] of Line Segment [tex]\( \overline{AB} \)[/tex]

Given points:
- [tex]\( A(2, 2) \)[/tex]
- [tex]\( B(3, 8) \)[/tex]

The formula for the slope [tex]\( m \)[/tex] of the line passing through two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substitute the given coordinates:
[tex]\[ m = \frac{8 - 2}{3 - 2} = \frac{6}{1} = 6 \][/tex]

So, the slope [tex]\( m \)[/tex] is [tex]\( 6 \)[/tex].

### Step 2: Calculate the Length of [tex]\( \overline{AB} \)[/tex] Using the Distance Formula

The distance formula between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Substitute the given coordinates:
[tex]\[ d = \sqrt{(3 - 2)^2 + (8 - 2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37} \][/tex]

So, the length of [tex]\( \overline{AB} \)[/tex] is [tex]\( \sqrt{37} \)[/tex].

### Step 3: Determine the Length of [tex]\( \overline{A'B'} \)[/tex] After Dilation

The problem specifies a dilation with a scale factor of 3.5. To find the new length of [tex]\( \overline{A'B'} \)[/tex], we multiply the original length [tex]\( \overline{AB} \)[/tex] by the scale factor:

[tex]\[ \text{Length of } \overline{A'B'} = \text{Length of } \overline{AB} \times \text{Scale Factor} \][/tex]

Substitute the known values:
[tex]\[ \text{Length of } \overline{A'B'} = \sqrt{37} \times 3.5 = 3.5\sqrt{37} \][/tex]

Thus, the length of [tex]\( \overline{A'B'} \)[/tex] is [tex]\( 3.5\sqrt{37} \)[/tex].

### Conclusion

From the above steps, we have determined that the slope [tex]\( m \)[/tex] of [tex]\( \overline{A'B'} \)[/tex] is [tex]\( 6 \)[/tex] and the length of [tex]\( \overline{A'B'} \)[/tex] is [tex]\( 3.5\sqrt{37} \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{B. \; m=6, \; A'B' = 3.5\sqrt{37}} \][/tex]
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