Select the coordinates A′ and B′ after dilation of the line segment AB with a scale factor of 4, centered at the origin where the endpoints are A(−2,−3) and B (−4,−5).
To find the coordinates of A' and B' after dilation with a scale factor of 4 centered at the origin, we first need to find the distance between the endpoints A and B.
Distance formula for two points (x1, y1) and (x2, y2) is given as:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Calculating the distance for AB:
d = sqrt((-4 - (-2))^2 + (-5 - (-3))^2)
d = sqrt((-2)^2 + (-2)^2)
d = sqrt(4 + 4)
d = sqrt(8)
d = 2sqrt(2)
Now, to dilate the line segment AB with a scale factor of 4, the distance needs to be multiplied by 4:
New distance = 2sqrt(2) * 4 = 8sqrt(2)
Next, we find the coordinates A' and B' by projecting the new distance along the same line as AB:
A' = (-2 * 4, -3 * 4) = (-8, -12)
B' = (-4 * 4, -5 * 4) = (-16, -20)
Therefore, the coordinates of A' and B' after dilation with a scale factor of 4 centered at the origin are A' (-8, -12) and B' (-16, -20) respectively.