Question

1. a line segment with endpoints p(1,2) and q(3,4) is dilated with a scale factor of \\(\frac{1}{2}\\). what are the coordinates of q?

a. (6,8)
b. (2,3)
c. (1.5,2)
d. (1.5,3)

Explanation:

Step1: Recall Dilation Formula

For a point \((x,y)\) dilated with scale factor \(k\) (assuming center at origin, as no center is given, standard dilation from origin), the new coordinates are \((kx, ky)\). Wait, but wait, maybe the scale factor is \( \frac{1}{2} \)? Wait, no, wait the problem says scale factor of \( \frac{1}{2} \)? Wait, no, wait the original problem: endpoints \(P(1,2)\) and \(Q(3,4)\), dilated with scale factor of \( \frac{1}{2} \)? Wait, no, maybe I misread. Wait, let's check again. Wait, the options: a. (6,8), b. (2,3), c. (1.5,2), d. (1.5,3). Wait, maybe the scale factor is \( \frac{1}{2} \)? Wait, no, wait \(Q\) is (3,4). If we dilate with scale factor \( \frac{1}{2} \)? No, (30.5, 40.5) is (1.5, 2), but option c is (1.5,2), but wait maybe the center is the midpoint? Wait, no, maybe the scale factor is \( \frac{1}{2} \) from the midpoint? Wait, no, let's re-express. Wait, maybe the scale factor is \( \frac{1}{2} \), but wait the midpoint of \(P(1,2)\) and \(Q(3,4)\) is \((\frac{1+3}{2}, \frac{2+4}{2})=(2,3)\). Wait, no, maybe the dilation is with scale factor \( \frac{1}{2} \) from the origin? No, (30.5,40.5)=(1.5,2), which is option c. Wait, but maybe the scale factor is 0.5? Wait, let's do the calculation.

Wait, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated with a scale factor of \( \frac{1}{2} \). What are the coordinates of \(Q'\)?"

So dilation formula: if center is origin, then \(Q'(k \times x, k \times y)\). So \(k = \frac{1}{2}\), so \(Q'(3 \times \frac{1}{2}, 4 \times \frac{1}{2}) = (1.5, 2)\)? But option c is (1.5,2), but wait option d is (1.5,3). Wait, maybe the center is \(P\)? No, that would be different. Wait, maybe I misread the scale factor. Wait, maybe the scale factor is \( \frac{1}{2} \) but center at \(P\)? No, let's check the options. Wait, option c is (1.5,2), which is (30.5,40.5). So that's dilation with scale factor 0.5 from origin. So:

Step2: Apply Dilation to \(Q(3,4)\)

Scale factor \(k = \frac{1}{2}\), so new \(x\)-coordinate: \(3 \times \frac{1}{2} = 1.5\), new \(y\)-coordinate: \(4 \times \frac{1}{2} = 2\). So \(Q'\) is (1.5, 2), which is option c. Wait, but wait option c is (1.5,2), yes.

Wait, but wait maybe the scale factor is \( \frac{1}{2} \) but the center is the midpoint? No, midpoint is (2,3). If we dilate \(Q(3,4)\) towards midpoint (2,3) with scale factor \( \frac{1}{2} \), the vector from midpoint to \(Q\) is (3-2,4-3)=(1,1). Scaling that vector by \( \frac{1}{2} \) gives (0.5,0.5), so new point is (2 + 0.5, 3 + 0.5)=(2.5,3.5), not an option. So probably center at origin, scale factor \( \frac{1}{2} \). So \(Q(3,4)\) dilated by \( \frac{1}{2} \) is (1.5,2), which is option c.

Wait, but the options: a. (6,8) is scaling by 2, b. (2,3) is midpoint, c. (1.5,2) is scaling by 0.5, d. (1.5,3) is (30.5, 40.75). So likely scale factor is \( \frac{1}{2} \), so \(Q'\) is (1.5,2), option c.

Step2: Calculate \(Q'\) Coordinates

Given \(Q(3,4)\) and scale factor \(k = \frac{1}{2}\) (assuming origin as center, standard dilation), new \(x = 3 \times \frac{1}{2} = 1.5\), new \(y = 4 \times \frac{1}{2} = 2\). So \(Q' = (1.5, 2)\), which matches option c.

Answer:

c. (1.5, 2)