Question

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

which description best fits the relationship between a preimage and its image in a dilation?
a. the image and preimage are congruent
b. the sides are the same length, and the angles are different
c. the angles are different, and the sides are proportional
d. the angles are congruent, and the sides are proportional

a square has vertices at $a(-2,-2)$, $b(-2,2)$, $c(2,2)$, and $d(2,-2)$. if it is dilated with a scale factor of 4 and the center of dilation is at $(0,0)$, what are the coordinates of $c$?
a. $(-8,8)$
b. $(8,8)$
c. $(-8,-8)$
d. $(8,-8)$

Explanation:

First Question:

Step1: Recall dilation formula

To dilate a point \((x,y)\) with a scale factor \(k\) centered at the origin (assuming center is origin here, as not specified otherwise), the new coordinates \((x',y')\) are given by \(x' = kx\), \(y' = ky\). For point \(Q(3,4)\) and scale factor \(\frac{1}{2}\).

Step2: Calculate new x-coordinate

\(x' = \frac{1}{2} \times 3 = 1.5\)

Step3: Calculate new y-coordinate

\(y' = \frac{1}{2} \times 4 = 2\)? Wait, no, wait. Wait, maybe the center is the same as the other point? Wait, no, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated". Usually, if not specified, dilation is centered at the origin? Wait, no, maybe the center is the same as the midpoint? Wait, no, the standard dilation when not specified center, sometimes it's centered at the origin, but maybe here, let's recheck. Wait, the coordinates of \(Q\) are \((3,4)\), scale factor \(\frac{1}{2}\). So applying dilation: \(x = 3 \times \frac{1}{2} = 1.5\), \(y = 4 \times \frac{1}{2} = 2\)? But option c is \((1.5,3)\). Wait, maybe the center is \(P\)? Wait, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated". So the vector from \(P\) to \(Q\) is \((3 - 1, 4 - 2) = (2,2)\). Dilation with scale factor \(\frac{1}{2}\) from \(P\) would be \(P + \frac{1}{2}(Q - P)\). So \(Q' = P + \frac{1}{2}(Q - P) = (1,2) + \frac{1}{2}(2,2) = (1 + 1, 2 + 1) = (2,3)\)? No, that's option d. Wait, I'm confused. Wait, maybe the center is the origin. Let's recalculate: \(Q(3,4)\) dilated by \(\frac{1}{2}\) is \((3 \times \frac{1}{2}, 4 \times \frac{1}{2}) = (1.5, 2)\)? But option a is \((1.5,2)\), option c is \((1.5,3)\). Wait, maybe the center is \(P\). Let's do that: The vector from \(P\) to \(Q\) is \((3 - 1, 4 - 2) = (2,2)\). Dilation with scale factor \(\frac{1}{2}\) from \(P\) would be \(P + \frac{1}{2}(Q - P) = (1,2) + (1,1) = (2,3)\)? No, that's (2,3), option d. Wait, maybe I made a mistake. Wait, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated with a scale factor of \(\frac{1}{2}\)". So the coordinates of \(Q'\) would be found by multiplying each coordinate of \(Q\) by the scale factor, assuming the center of dilation is the origin. So \(3 \times \frac{1}{2} = 1.5\), \(4 \times \frac{1}{2} = 2\), so \((1.5, 2)\), which is option a? But wait, maybe the center is \(P\). Let's check the distance between \(P\) and \(Q\): \(\sqrt{(3 - 1)^2 + (4 - 2)^2} = \sqrt{4 + 4} = \sqrt{8}\). After dilation with scale factor \(\frac{1}{2}\), the distance should be \(\sqrt{2}\). The distance between \(P(1,2)\) and \(Q'(1.5,2)\) is \(0.5\), which is not \(\sqrt{2}\). Wait, no, if center is origin, distance from origin to \(Q\) is \(\sqrt{3^2 + 4^2} = 5\), after dilation, distance is \(2.5\). Distance from origin to \(Q'(1.5,2)\) is \(\sqrt{1.5^2 + 2^2} = \sqrt{2.25 + 4} = \sqrt{6.25} = 2.5\), which matches. So \(Q'\) is \((1.5,2)\), option a? Wait, but the options are a. (1.5,2), b. (6,8), c. (1.5,3), d. (2,3). Wait, maybe the center is \(P\). Let's recalculate: \(Q - P = (3 - 1, 4 - 2) = (2,2)\). Dilation with scale factor \(\frac{1}{2}\) gives \(\frac{1}{2}(2,2) = (1,1)\). Then \(Q' = P + (1,1) = (1 + 1, 2 + 1) = (2,3)\), which is option d. But why the discrepancy? Maybe the problem assumes the center of dilation is the origin. Wait, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated". The standard dilation when not specified center is origin. So \(Q(3,4)\) dilated by \(\frac{1}{2}\) is \((1.5, 2)\), option a. But let's check the answer opti…

Dilation is a transformation that changes the size of a figure but not its shape. In dilation, the angles of the image and preimage remain congruent (same measure), and the sides are proportional (scaled by the scale factor). Option a is incorrect because congruent means same size and shape, but dilation changes size. Option b is incorrect because sides are not same length (they are scaled) and angles are same. Option c is incorrect because angles are congruent, not different. Option d is correct as angles are congruent and sides are proportional.

Step1: Recall dilation formula

To dilate a point \((x,y)\) with a scale factor \(k\) centered at the origin \((0,0)\), the new coordinates \((x',y')\) are given by \(x' = kx\), \(y' = ky\). For point \(C(2,2)\) and scale factor \(4\).

Step2: Calculate new x-coordinate

\(x' = 4 \times 2 = 8\)

Step3: Calculate new y-coordinate

\(y' = 4 \times 2 = 8\)
So the coordinates of \(C'\) are \((8,8)\).

Answer:

d. The angles are congruent, and the sides are proportional

Third Question: