Question

in the diagram, zy = wx = 2\sqrt{10}. what is the perimeter of rectangle wxyz? \bigcirc \sqrt{10} units \bigcirc 2\sqrt{10} units \bigcirc 6\sqrt{10} units \bigcirc 8\sqrt{10} units (diagram shows coordinates: w(-6,6), x(-4,0), y(-7,-1), z(-9,5) on a grid)

Explanation:

Step1: Find the length of WY or XZ (another side of the rectangle)

We can use the distance formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \): \( d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Let's take points \( W(-6,6) \) and \( Y(-7,-1) \).
\( d=\sqrt{(-7 - (-6))^2+(-1 - 6)^2}=\sqrt{(-1)^2+(-7)^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2} \)? Wait, no, wait. Wait, actually, in a rectangle, opposite sides are equal. Wait, we know \( ZY = WX=2\sqrt{10} \), so we need to find the length of \( WZ \) or \( XY \). Let's take points \( Z(-9,5) \) and \( W(-6,6) \).
\( d=\sqrt{(-6 - (-9))^2+(6 - 5)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9 + 1}=\sqrt{10} \). Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's check points \( W(-6,6) \) and \( X(-4,0) \). Wait, no, the rectangle is WXYZ, so the sides are WX, XY, YZ, ZW. Wait, we know WX and ZY are equal (given as \( 2\sqrt{10} \)). Now we need to find the length of WZ or XY. Let's take points \( W(-6,6) \) and \( Z(-9,5) \). The distance between W and Z: \( \Delta x=-9 - (-6)=-3 \), \( \Delta y = 5 - 6=-1 \). So distance \( WZ=\sqrt{(-3)^2+(-1)^2}=\sqrt{9 + 1}=\sqrt{10} \). Wait, but that would mean the sides are \( 2\sqrt{10} \) and \( \sqrt{10} \)? No, wait, maybe I took the wrong points. Wait, let's take points \( Y(-7,-1) \) and \( X(-4,0) \). Distance between Y and X: \( \Delta x=-4 - (-7)=3 \), \( \Delta y=0 - (-1)=1 \). So distance \( XY=\sqrt{3^2 + 1^2}=\sqrt{9 + 1}=\sqrt{10} \). Ah, so the rectangle has length \( 2\sqrt{10} \) and width \( \sqrt{10} \)? Wait, no, wait, perimeter of a rectangle is \( 2(l + w) \). Wait, but let's check again. Wait, the given sides are \( ZY = WX = 2\sqrt{10} \), so those are two opposite sides. Then the other two opposite sides (WZ and XY) should be equal. Let's calculate WZ: points W(-6,6) and Z(-9,5). \( \Delta x=-9 - (-6)=-3 \), \( \Delta y=5 - 6=-1 \). So distance \( \sqrt{(-3)^2+(-1)^2}=\sqrt{9 + 1}=\sqrt{10} \). So the length is \( 2\sqrt{10} \), width is \( \sqrt{10} \). Then perimeter is \( 2(2\sqrt{10}+\sqrt{10})=2(3\sqrt{10}) = 6\sqrt{10} \)? Wait, no, wait, that can't be. Wait, maybe I messed up the sides. Wait, let's check the coordinates again. W(-6,6), X(-4,0), Y(-7,-1), Z(-9,5). So WX: from (-6,6) to (-4,0). \( \Delta x=-4 - (-6)=2 \), \( \Delta y=0 - 6=-6 \). So distance \( \sqrt{2^2+(-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10} \). Ah! There we go, I made a mistake earlier. So WX is from (-6,6) to (-4,0): \( \Delta x = -4 - (-6)=2 \), \( \Delta y=0 - 6=-6 \). So distance \( \sqrt{(2)^2+(-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10} \), which matches the given. Now let's find XY: from (-4,0) to (-7,-1). \( \Delta x=-7 - (-4)=-3 \), \( \Delta y=-1 - 0=-1 \). Wait, no, that's not right. Wait, no, in a rectangle, adjacent sides are perpendicular. Wait, WX is from W(-6,6) to X(-4,0), vector (2, -6). Then XY should be a vector perpendicular to (2, -6), so the dot product should be zero. Let's take X(-4,0) to Y(-7,-1): vector (-3, -1). Dot product with (2, -6) is (2)(-3) + (-6)(-1) = -6 + 6 = 0. So they are perpendicular. Now distance XY: \( \sqrt{(-3)^2+(-1)^2}=\sqrt{9 + 1}=\sqrt{10} \)? No, wait, no: \( (-3)^2 + (-1)^2 = 9 + 1 = 10 \), so \( \sqrt{10} \). Wait, but WX is \( 2\sqrt{10} \), XY is \( \sqrt{10} \). Then perimeter is \( 2(WX + XY)=2(2\sqrt{10}+\sqrt{10})=2(3\sqrt{10})=6\sqrt{10} \). Wait, but let's check ZY: from Z(-9,5) to Y(-7,-1). \( \Delta x=-7 - (-9)=2 \), \( \Delta y=-1 - 5=-6 \). So distance \( \sqrt{2^2+(-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10} \), which matches. And WZ: from W(-6,6) to Z(-9,5). \( \Delta x=-9 - (-6)=-3 \), \( \Delta y=5 - 6=-1 \). Distance \( \sqrt{(-3)^2+(-1)^2}=\sqrt{10} \), which is equal to XY. So the sides are \( 2\sqrt{10} \) and \( \sqrt{10} \). Then perimeter is \( 2(2\sqrt{10}+\sqrt{10})=2(3\sqrt{10})=6\sqrt{10} \). Wait, but the options include \( 6\sqrt{10} \) units. Wait, but let's recheck the distance between W(-6,6) and X(-4,0): \( \sqrt{(-4 - (-6))^2 + (0 - 6)^2}=\sqrt{(2)^2 + (-6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10} \), correct. Distance between X(-4,0) and Y(-7,-1): \( \sqrt{(-7 - (-4))^2 + (-1 - 0)^2}=\sqrt{(-3)^2 + (-1)^2}=\sqrt{9 + 1}=\sqrt{10} \), correct. So perimeter is \( 2*(2\sqrt{10}+\sqrt{10})=2*3\sqrt{10}=6\sqrt{10} \).

Step2: Calculate the perimeter

Perimeter of a rectangle is \( P = 2(l + w) \), where \( l = 2\sqrt{10} \) and \( w=\sqrt{10} \).
\( P = 2(2\sqrt{10}+\sqrt{10}) = 2(3\sqrt{10}) = 6\sqrt{10} \)

Answer:

\( 6\sqrt{10} \) units (corresponding to the option "6√10 units")