Question

select the correct answer from each drop - down menu. polygon pqrs, shown in the figure, is dilated by a scale factor of 0.5 with the origin as the center of dilation, resulting in polygon pqrs. the slope of overline{pq} is \boxed{} the length of overline{pq} is approximately \boxed{}

Explanation:

Step1: Find coordinates of P and Q

From the graph, assume P is at \((9, 10)\) and Q is at \((6, 5)\) (need to check grid: x and y axes, P seems at x=9, y=10; Q at x=6, y=5).

Step2: Calculate slope of PQ

Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For P\((9,10)\) and Q\((6,5)\), \(m = \frac{5 - 10}{6 - 9} = \frac{-5}{-3} = \frac{5}{3}\). Dilation preserves slope, so slope of \(P'Q'\) is also \(\frac{5}{3}\) (or check if coordinates are correct: maybe P is (9,10), Q is (6,5)? Wait, maybe I mixed x and y. Wait, x-axis is horizontal (bottom), y-axis vertical (left). So x increases right, y increases up. So P: x=9, y=10? Wait, no, the x-axis is labeled 0 to 20 at the bottom, y-axis 0 to 20 on the left. So P is at (9,10)? Wait, Q is at (6,5)? Wait, let's recheck: Q is at x=6, y=14? Wait, maybe I misread. Wait, the y-axis is on the left, so the vertical axis is y, horizontal is x. So Q: x=6, y=14? P: x=9, y=10? Wait, no, the blue polygon: P, Q, R, S. Let's find coordinates properly.

Wait, let's take Q: looking at the grid, Q is at (6, 14)? Wait, no, the y-axis is on the left, so the top is y=20, bottom y=0. x-axis: left x=0, right x=20. So Q: x=6, y=14? P: x=9, y=10? Wait, no, maybe P is (9,10), Q is (6,14)? Wait, the blue line from P to Q: let's count the grid squares. From P to Q: horizontal change (x): 6 - 9 = -3, vertical change (y): 14 - 10 = 4? Wait, maybe I had y reversed. Let's correct: y-axis is vertical, so higher y is up. So Q is at (6,14), P at (9,10). Then slope: \(\frac{10 - 14}{9 - 6} = \frac{-4}{3} = -\frac{4}{3}\)? No, that doesn't make sense. Wait, maybe the coordinates are P(9,10) and Q(6,14)? Wait, the graph is a bit unclear, but let's assume P is (9,10) and Q is (6,14). Then horizontal difference: 9 - 6 = 3, vertical difference: 10 - 14 = -4. So slope is \(\frac{-4}{3}\)? No, maybe I got x and y wrong. Wait, the x-axis is the horizontal axis at the bottom, so x increases to the right, y increases upward (left axis). So Q is at (6,14) (x=6, y=14), P is at (9,10) (x=9, y=10). Then slope of PQ: \(\frac{10 - 14}{9 - 6} = \frac{-4}{3} = -\frac{4}{3}\)? No, that's negative. Wait, maybe Q is at (6,5) and P at (9,10). Then vertical change: 5 - 10 = -5, horizontal: 6 - 9 = -3, slope \(\frac{5}{3}\). That's positive. Let's check the length of PQ. Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For P(9,10) and Q(6,5): \(d = \sqrt{(6 - 9)^2 + (5 - 10)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83\). Dilation by scale factor 0.5: new length is \(0.5 \times \sqrt{34} \approx 2.915\), approximately 2.92 or 3. But maybe the coordinates are different. Alternatively, if P is (8,10) and Q is (5,5), then slope is \(\frac{5-10}{5-8} = \frac{-5}{-3} = \frac{5}{3}\), length \(\sqrt{(5-8)^2 + (5-10)^2} = \sqrt{9 +25}=\sqrt{34}\approx5.83\), dilated length \(\approx2.91\).

But the key point: dilation preserves slope, so slope of \(P'Q'\) is equal to slope of PQ.

Step1 (correct): Find P and Q coordinates

Let's correctly identify coordinates:
- P: x=9, y=10 (since x-axis: 9 units right, y-axis: 10 units up)
- Q: x=6, y=5 (x=6 right, y=5 up)

Step2: Slope of PQ

Slope \(m = \frac{y_Q - y_P}{x_Q - x_P} = \frac{5 - 10}{6 - 9} = \frac{-5}{-3} = \frac{5}{3}\). Dilation (scaling from origin) preserves the slope, so slope of \(P'Q'\) is \(\frac{5}{3}\) (or \(1.\overline{6}\)).

Step3: Length of PQ

Using distance formula: \(PQ = \sqrt{(x_Q - x_P)^2 + (y_Q - y_P)^2} = \sqrt{(6 - 9)^2 + (5 - 10)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83\). Dilation by scale factor 0.5: \(P'Q' = 0.5 \times \sqrt{34} \approx 2.915 \approx 2.92\) (or if we take exact \(\sqrt{34} \approx 5.83\), half is ~2.91).

But maybe the coordinates are P(9,10) and Q(6,14). Let's recalculate: \(x_Q - x_P = 6 - 9 = -3\), \(y_Q - y_P = 14 - 10 = 4\). Slope: \(\frac{4}{-3} = -\frac{4}{3}\). Length: \(\sqrt{(-3)^2 + 4^2} = 5\). Then dilated length: \(5 \times 0.5 = 2.5\). Ah, maybe that's better. So if P is (9,10) and Q is (6,14), then horizontal change: -3, vertical change: +4. Slope: \(\frac{4}{-3} = -\frac{4}{3}\). Length: 5. Then dilated length: 2.5.

Wait, the graph: let's count the grid squares. From P to Q: how many horizontal and vertical? If P is at (9,10) (x=9, y=10) and Q at (6,14) (x=6, y=14), then moving from P to Q: left 3 (x-3), up 4 (y+4). So slope is \(\frac{4}{-3} = -\frac{4}{3}\), length 5 (3-4-5 triangle). Then dilation by 0.5: length becomes 2.5, slope remains \(-\frac{4}{3}\) (or \(\frac{4}{-3}\)).

This is confusing, but the key is: dilation preserves slope (since it's a similarity transformation), so slope of \(P'Q'\) is equal to slope of PQ. For length, dilation scales length by the scale factor (0.5), so length of \(P'Q'\) is 0.5 times length of PQ.

Assuming PQ has length 5 (3-4-5 triangle), then \(P'Q'\) length is 2.5. Slope: \(-\frac{4}{3}\) or \(\frac{5}{3}\) depending on coordinates. But let's proceed with the correct coordinates.

Final: Slope of \(P'Q'\) is equal to slope of PQ (dilation preserves slope), length is 0.5 * length of PQ.

Answer:

Slope: \(\boldsymbol{\frac{5}{3}}\) (or \(-\frac{4}{3}\) depending on coordinates), Length: \(\boldsymbol{\approx 2.92}\) (or 2.5 if PQ is 5). But assuming PQ length is 5 (3-4-5), then length is 2.5, slope \(-\frac{4}{3}\).