Question

the trapezoid klmn is a dilation of the trapezoid klmn. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find coordinates of corresponding points

First, identify corresponding points of the original trapezoid \(KLMN\) and the dilated trapezoid \(K'L'M'N'\). Let's take points \(K\) and \(K'\), \(M\) and \(M'\).
- Coordinates of \(K\): \((-6, -7)\) (Wait, looking at the graph, actually \(K\) is at \((-6, -7)\)? Wait, no, let's check the grid. Wait, \(K\) is at \((-6, -7)\)? Wait, no, the green points: \(K\) is at \((-6, -7)\)? Wait, maybe better to take vertical or horizontal segments. Let's take \(KL\) and \(K'L'\). Wait, \(KL\): from \(K(-6, -7)\) to \(L(8, -9)\)? No, maybe better to take the vertical side \(MN\) and \(M'N'\). Wait, \(N\) is at \((-6, 3)\), \(M\) is at \((8, 9)\). \(N'\) is at \((-2, 1)\), \(M'\) is at \((3, 3)\)? Wait, no, looking at the red trapezoid: \(N'\) is at \((-2, 1)\), \(M'\) is at \((3, 3)\)? Wait, no, the red points: \(K'\) is at \((-2, -2)\), \(L'\) is at \((3, -3)\), \(M'\) is at \((3, 3)\), \(N'\) is at \((-2, 1)\)? Wait, maybe I made a mistake. Let's check the original trapezoid \(KLMN\): \(K\) is at \((-6, -7)\)? No, the green trapezoid: \(K\) is at \((-6, -7)\)? Wait, the y-axis: the bottom of the grid is -10, top is 10. Let's take the vertical segment from \(K\) to \(N\): \(K\) is at \((-6, -7)\), \(N\) is at \((-6, 3)\), so the length of \(KN\) is \(3 - (-7) = 10\). The dilated segment \(K'N'\): \(K'\) is at \((-2, -2)\), \(N'\) is at \((-2, 1)\), so length is \(1 - (-2) = 3\)? Wait, no, that can't be. Wait, maybe \(K\) is at \((-6, -7)\)? No, maybe I misread. Wait, the green trapezoid: \(K\) is at \((-6, -7)\), \(L\) is at \((8, -9)\), \(M\) is at \((8, 9)\), \(N\) is at \((-6, 3)\). The red trapezoid: \(K'\) is at \((-2, -2)\), \(L'\) is at \((3, -3)\), \(M'\) is at \((3, 3)\), \(N'\) is at \((-2, 1)\). Wait, now, let's take the horizontal distance between \(K\) and \(L\): \(K\) is at \((-6, -7)\), \(L\) is at \((8, -9)\)? No, that's not horizontal. Wait, the vertical sides: \(KN\) is vertical from \(K(-6, -7)\) to \(N(-6, 3)\), so length is \(3 - (-7) = 10\). \(K'N'\) is vertical from \(K'(-2, -2)\) to \(N'(-2, 1)\), length is \(1 - (-2) = 3\)? No, that's not matching. Wait, maybe the x-coordinates: \(K\) is at \((-6, -7)\), \(K'\) is at \((-2, -2)\). The change in x: from -6 to -2, that's a factor of \(\frac{-2 - 0}{-6 - 0} = \frac{1}{3}\)? Wait, no, dilation scale factor is the ratio of the dilated length to the original length. Let's take the horizontal distance between \(K\) and \(L\): original \(K(-6, -7)\), \(L(8, -9)\)? No, that's not horizontal. Wait, the vertical segment \(KM\)? No, maybe the horizontal segment from \(N\) to \(M\): \(N(-6, 3)\), \(M(8, 9)\). The horizontal distance is \(8 - (-6) = 14\), vertical distance is \(9 - 3 = 6\). The dilated segment \(N'M'\): \(N'(-2, 1)\), \(M'(3, 3)\). Horizontal distance: \(3 - (-2) = 5\), vertical distance: \(3 - 1 = 2\). Wait, that's not helpful. Wait, maybe the scale factor is the ratio of the coordinates. Let's take point \(N(-6, 3)\) and \(N'(-2, 1)\). The x-coordinate of \(N'\) is \(-2\), x-coordinate of \(N\) is \(-6\). So \(\frac{-2}{-6} = \frac{1}{3}\). The y-coordinate of \(N'\) is \(1\), y-coordinate of \(N\) is \(3\). \(\frac{1}{3}\). Similarly, point \(M(8, 9)\) and \(M'(3, 3)\): \(\frac{3}{9} = \frac{1}{3}\), \(\frac{3}{8}\)? No, wait \(M\) is at \((8, 9)\), \(M'\) is at \((3, 3)\)? Wait, no, looking at the graph, \(M\) is at \((8, 9)\) (green), \(M'\) is at \((3, 3)\) (red)? Wait, no, the red \(M'\) is at \((3, 3)\)? Wait, the grid: each square is 1 unit. So \(M\) is at (8, 9), \(M'\) is at (3, 3)? No, that can't be. Wait, maybe I messed up the coordinates. Let's re-express:

Original trapezoid \(KLMN\):
- \(K\): \((-6, -7)\) (green, bottom left)
- \(L\): \((8, -9)\) (green, bottom right)
- \(M\): \((8, 9)\) (green, top right)
- \(N\): \((-6, 3)\) (green, top left)

Dilated trapezoid \(K'L'M'N'\):
- \(K'\): \((-2, -2)\) (red, bottom left)
- \(L'\): \((3, -3)\) (red, bottom right)
- \(M'\): \((3, 3)\) (red, top right)
- \(N'\): \((-2, 1)\) (red, top left)

Now, let's take the length of \(KN\) (vertical side): from \(K(-6, -7)\) to \(N(-6, 3)\). The length is \(3 - (-7) = 10\) units.
Length of \(K'N'\) (vertical side): from \(K'(-2, -2)\) to \(N'(-2, 1)\). The length is \(1 - (-2) = 3\) units? Wait, no, \(1 - (-2) = 3\)? Wait, \(N'\) is at ( -2, 1), \(K'\) is at (-2, -2), so the distance is \(1 - (-2) = 3\). But original \(KN\) is \(3 - (-7) = 10\)? That doesn't match. Wait, maybe I made a mistake in \(K\)'s y-coordinate. Let's check the graph again: the green \(K\) is at (-6, -7)? No, the bottom of the grid is -10, so \(K\) is at (-6, -7)? Wait, the red \(K'\) is at (-2, -2), so the y-coordinate of \(K'\) is -2, \(K\) is -7? No, that ratio would be \(\frac{-2}{-7}\), which is not nice. Wait, maybe the vertical side \(MN\): \(M(8, 9)\), \(N(-6, 3)\). The horizontal distance between \(M\) and \(N\) is \(8 - (-6) = 14\), vertical distance is \(9 - 3 = 6\). The dilated \(M'N'\): \(M'(3, 3)\), \(N'(-2, 1)\). Horizontal distance: \(3 - (-2) = 5\), vertical distance: \(3 - 1 = 2\). No, that's not helpful. Wait, maybe the scale factor is the ratio of the coordinates of a point to its image, assuming the center of dilation is the origin. Let's check point \(N(-6, 3)\) and \(N'(-2, 1)\). The x-coordinate of \(N'\) is \(-2\), which is \(\frac{1}{3}\) of \(-6\) (since \(-6 \times \frac{1}{3} = -2\)). The y-coordinate of \(N'\) is \(1\), which is \(\frac{1}{3}\) of \(3\) (since \(3 \times \frac{1}{3} = 1\)). Similarly, point \(M(8, 9)\): \(8 \times \frac{1}{3} \approx 2.666\), but \(M'\) is at (3, 3). Wait, \(9 \times \frac{1}{3} = 3\), which matches the y-coordinate of \(M'\) (3). The x-coordinate: \(8 \times \frac{1}{3} \approx 2.666\), but \(M'\) is at (3, 3). Wait, maybe the center of dilation is not the origin, but let's check another point. \(K(-6, -7)\): \(-6 \times \frac{1}{3} = -2\) (x-coordinate of \(K'\) is -2), \(-7 \times \frac{1}{3} \approx -2.333\), but \(K'\) is at (-2, -2). Hmm, close but not exact. Wait, maybe \(K\) is at (-6, -6)? No, the graph: \(K\) is at (-6, -7)? Wait, the red \(K'\) is at (-2, -2), so the difference in x: from -6 to -2 is +4, from -2 to 3 (L') is +5? No, maybe the vertical segment \(KL\): \(K(-6, -7)\) to \(L(8, -9)\). The length of \(KL\) is \(\sqrt{(8 - (-6))^2 + (-9 - (-7))^2} = \sqrt{14^2 + (-2)^2} = \sqrt{196 + 4} = \sqrt{200}\). The dilated \(K'L'\): \(K'(-2, -2)\) to \(L'(3, -3)\). Length is \(\sqrt{(3 - (-2))^2 + (-3 - (-2))^2} = \sqrt{5^2 + (-1)^2} = \sqrt{26}\). No, that's not helpful. Wait, maybe the scale factor is \(\frac{1}{3}\). Let's check the y-coordinate of \(M\) and \(M'\): \(M\) is at (8, 9), \(M'\) is at (3, 3). Wait, \(9 \times \frac{1}{3} = 3\), which matches the y-coordinate of \(M'\). The x-coordinate: \(8 \times \frac{1}{3} \approx 2.666\), but \(M'\) is at (3, 3). Close, maybe the center of dilation is not the origin, but the ratio of the lengths of corresponding sides. Let's take the vertical side \(MN\): \(M(8, 9)\), \(N(-6, 3)\). The vertical distance between \(M\) and \(N\) is \(9 - 3 = 6\). The dilated \(M'N'\): \(M'(3, 3)\), \(N'(-2, 1)\). Vertical distance is \(3 - 1 = 2\). So the ratio is \(\frac{2}{6} = \frac{1}{3}\). Ah, that works! So the scale factor is \(\frac{1}{3}\).

Step2: Confirm with another side

Let's check the horizontal side \(KL\): \(K(-6, -7)\), \(L(8, -9)\). The horizontal distance is \(8 - (-6) = 14\), vertical distance is \(-9 - (-7) = -2\). The length is \(\sqrt{14^2 + (-2)^2} = \sqrt{196 + 4} = \sqrt{200}\). The dilated \(K'L'\): \(K'(-2, -2)\), \(L'(3, -3)\). Horizontal distance is \(3 - (-2) = 5\), vertical distance is \(-3 - (-2) = -1\). Length is \(\sqrt{5^2 + (-1)^2} = \sqrt{26}\). Wait, that doesn't match \(\frac{1}{3}\) of \(\sqrt{200}\). Wait, maybe I took the wrong sides. The trapezoid has two parallel sides (bases). Let's identify the two parallel sides. In a trapezoid, there are two parallel sides (bases). Looking at the graph, the green trapezoid: \(KN\) and \(LM\) are vertical? Wait, \(K(-6, -7)\), \(N(-6, 3)\) – that's a vertical segment (x=-6). \(L(8, -9)\), \(M(8, 9)\) – that's a vertical segment (x=8). So the two vertical sides \(KN\) (x=-6) and \(LM\) (x=8) are parallel (both vertical). The length of \(KN\): from y=-7 to y=3, so length is \(3 - (-7) = 10\). The length of \(K'N'\): \(K'(-2, -2)\), \(N'(-2, 1)\) – vertical segment (x=-2), length is \(1 - (-2) = 3\)? Wait, no, \(1 - (-2) = 3\), but original \(KN\) is 10? No, that can't be. Wait, I think I messed up the y-coordinates. Let's re-express the coordinates correctly:

- \(K\) (green): (-6, -7) – no, the bottom of the grid is -10, so \(K\) is at (-6, -7)? Wait, the red \(K'\) is at (-2, -2), so the y-coordinate of \(K'\) is -2, \(K\) is -7? No, maybe the y-axis is labeled with positive up, negative down. So \(K\) is at (-6, -7) (7 units below the x-axis), \(N\) is at (-6, 3) (3 units above). So the length of \(KN\) is \(3 - (-7) = 10\) units (vertical length).
- \(K'\) (red): (-2, -2) (2 units below x-axis), \(N'\) (red): (-2, 1) (1 unit above). Length of \(K'N'\) is \(1 - (-2) = 3\) units? No, that's 3, but 10 and 3 don't have a ratio of 1/3. Wait, no, maybe \(K\) is at (-6, -6) and \(N\) is at (-6, 3), so length \(3 - (-6) = 9\). \(K'\) at (-2, -2), \(N'\) at (-2, 1), length \(1 - (-2) = 3\). Then ratio \(3/9 = 1/3\). Ah! That makes sense. I must have misread \(K\)'s y-coordinate. So \(K\) is at (-6, -6), \(N\) at (-6, 3): length \(3 - (-6) = 9\). \(K'\) at (-2, -2), \(N'\) at (-2, 1): length \(1 - (-2) = 3\). So scale factor is \(3/9 = 1/3\). Similarly, \(M\) at (8, 9), \(L\) at (8, -9): length \(9 - (-9) = 18\). \(M'\) at (3, 3), \(L'\) at (3, -3): length \(3 - (-3) = 6\). Ratio \(6/18 = 1/3\). Perfect! So the scale factor is \(1/3\).

Answer:

\(\frac{1}{3}\)