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The endpoints of line segment AB are A (9,4) and B (5,-4). Then endpoints of its image after dilation are A' (6,3) and B' (5,-4). Find the scale factor and explain

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pmayl
To find our scale factor, let's determine the length of AB, and compare it to the length of A'B'.

The length of AB is the distance from (9,4) to (5,-4). 
Let's apply the distance formula:
AB = √ ((9-5)² + (4+4)²)
AB = √ (4² + 8²)
AB = √(16 + 64)
AB = √ (80)
AB = 4 √5
The length of AB is 4 √5.
Now for A'B':
A'B' = √ ((6-5)² + (3+4)²)
A'B' = √( 1 + 49)
A'B' = √ 50
A'B' = 5 √2
To find our scale factor k:
A'B' = k AB
5 √2 = 4 √5 * k Divide both sides by 4√5
5 √2 / (4 √5) = k                 To simplify, multiply the left side by (√5 / √5)
(5 *√5 * √2) / (4 *5) = k      the 5's in the numerator and denominator cancel
√5 * √2 / 4 = k                    
√10 / 4 = k

To confirm, make sure AB * k = A'B', using the values that we've calculated.

jollyfamily

Answer:To find the image of B, first find the scale factor for the dilation. The scale factor should be greater than 1 because the image of A is farther from the origin than A. Divide the coordinates of the image of A by the coordinates of A: –6/–4 = 3/2 and 9/6 = 3/2, so the scale factor is 3/2. Now, apply the dilation to B by multiplying the coordinates by 3/2 to get ((3/2)(1), (3/2)(4)), or (3/2, 6).

Hope I helped

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