Under a dilation with center at the origin and scale factor k, a point (x,y) maps to which coordinates?

Under a dilation with center at the origin and scale factor k, a point (x,y) maps to which coordinates?

• A. (x/k, y/k)
• B. (kx, ky)
• C. (x+y, y+x)
• D. (kx, y)

Answer: (B)

When you dilate with the center at the origin, you scale every distance from the origin by the factor k, keeping the direction from the origin the same. That means the position vector (x, y) becomes k times every component, giving (kx, ky). For example, if k = 2, (3, 4) goes to (6, 8); if k = 1/2, it goes to (1.5, 2). Negative k would flip the point to the opposite ray and scale its distance. The other possibilities don’t keep the same direction and scale both coordinates by the same factor, so they don’t represent the dilation about the origin.

When you dilate with the center at the origin, you scale every distance from the origin by the factor k, keeping the direction from the origin the same. That means the position vector (x, y) becomes k times every component, giving (kx, ky). For example, if k = 2, (3, 4) goes to (6, 8); if k = 1/2, it goes to (1.5, 2). Negative k would flip the point to the opposite ray and scale its distance. The other possibilities don’t keep the same direction and scale both coordinates by the same factor, so they don’t represent the dilation about the origin.