Triangle ABC with vertices at A(3, 2), B(2, −1), C(−2, 1) is dilated using a scale factor of 1.5 to create triangle A′B′C′. Determine the vertex of point A′.
A′(−4.5, 2)
A′(3, 3)
A′(4.5, 2)
A′(4.5, 3)
To find the vertex of point A′ after dilation by a scale factor of 1.5, we first need to find the distance between the original point A(3, 2) and the center of dilation.
The center of dilation is the origin (0,0) since the scale factor is 1.5.
The distance between A and the origin can be found using the distance formula:
d = sqrt((3-0)^2 + (2-0)^2)
d = sqrt(9 + 4)
d = sqrt(13)
To find the coordinates of A′, we need to multiply the distance between A and the origin by the scale factor and then multiply the direction vector by the ratio of the new distance to the old distance.
Since the scale factor is 1.5, the new distance for A′ would be:
d′ = 1.5 * sqrt(13)
The direction vector can be found by dividing the difference in coordinates by the distance between A and the origin:
direction vector = ((0-3) / sqrt(13), (0-2) / sqrt(13))
direction vector = (-3/sqrt(13), -2/sqrt(13))
Now we can find the coordinates of A′:
A′ = (0,0) + d′ * direction vector
A′ = (0,0) + 1.5 * sqrt(13) * (-3/sqrt(13), -2/sqrt(13))
A′ = (-4.5, -3)
Therefore, the vertex of point A′ after dilation is A′(-4.5, 2). So the correct answer is A′(-4.5, 2).