Question

The points A(0,2), B(4,0), and C(3,4) are plotted below. Find and plot their image points after a dilation by a factor of 2 centered at the origin, O.

Options:
A. A'(0,4), B'(8,0), C'(6,8)
B. A'(0,1), B'(2,0), C'(1.5,2)
C. A'(0,6), B'(12,0), C'(9,12)
D. A'(0,8), B'(16,0), C'(12,16)
How can you verify that C' is twice the distance from the origin as C?
Options:
A. Calculate the Euclidean distance from the origin to C and C' and compare them.
B. Measure the angle between OC and OC', it should be 90 degrees.
C. Use a protractor to measure the angle between C and C' and compare it to the angle between O and C.
D. Verify that the x-coordinate of C' is double the x-coordinate of C and the y-coordinate of C' is double the y-coordinate of C.

Answer 1

The image points after dilation by a factor of 2 are A'(0,4), B'(8,0), C'(6,8), and C' is twice the distance from the origin as C because both the x-coordinate and y-coordinate of C' are double those of C.

Step-by-step explanation:

To find the image points after a dilation by a factor of 2 centered at the origin, you need to multiply each coordinate of the original points by the dilation factor. For point A(0,2), multiplying each coordinate by 2 gives A'(0,4). Similarly, B(4,0) becomes B'(8,0), and C(3,4) becomes C'(6,8). Therefore, the correct answer is A. A'(0,4), B'(8,0), C'(6,8). To verify that C' is twice the distance from the origin as C, you calculate the Euclidean distance from the origin to C and C'. The distance formula for a point (x, y) from the origin (0,0) is √(x²+y²). So for point C(3,4), the distance is √(3²+4²)=5, and for C'(6,8) it is √(6²+8²)=10, which is twice the distance, verifying that the dilation has indeed doubled the distance from the origin. Therefore, the correct option is D. Verify that both the x-coordinate and y-coordinate of C' are double those of C.