Question

in the diagram, zy = wx = 2√10. what is the perimeter of rectangle wxyz?
√10 units
2√10 units
6√10 units
8√10 units

Explanation:

Step1: Calculate the length of \( WY \)

Use the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \).
For \( W(-6,6) \) and \( Y(-7,-1) \), \( x_1=-6,y_1 = 6,x_2=-7,y_2=-1 \).
\( WY=\sqrt{(-7 + 6)^2+(-1 - 6)^2}=\sqrt{(-1)^2+(-7)^2}=\sqrt{1 + 49}=\sqrt{50}=5\sqrt{2} \) (This step is wrong, let's use the property of rectangle. In a rectangle, opposite sides are equal. Since \( ZY = WX=2\sqrt{10} \), and for a rectangle \( P = 2(l + w) \). We can also use the fact that if we consider the movement from \( W(-6,6) \) to \( X(-4,0) \): horizontal change \( \Delta x=-4+6 = 2 \), vertical change \( \Delta y=0 - 6=-6 \). The length \( WX=\sqrt{( - 4+6)^2+(0 - 6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10} \). From \( W(-6,6) \) to \( Z(-9,5) \): horizontal change \( \Delta x=-9 + 6=-3 \), vertical change \( \Delta y=5 - 6=-1 \). The length \( WZ=\sqrt{(-9 + 6)^2+(5 - 6)^2}=\sqrt{9+1}=\sqrt{10} \). But wait, no. Wait, in a rectangle \( P=2(WX+WZ) \). Wait, another approach: Since it's a rectangle, perimeter \( P = 2(WX+ZY)\) (opposite sides equal). Wait, no, \( WX = ZY \), \( WZ=XY \). Let's use the slope - free method.
We know that in a rectangle \( P=2(a + b)\) where \( a\) and \( b\) are the lengths of adjacent sides.
We can find the length of another side. Let's use the fact that if we consider the vector from \( W(-6,6) \) to \( Z(-9,5) \): \( d=\sqrt{(-9 + 6)^2+(5 - 6)^2}=\sqrt{9 + 1}=\sqrt{10}\). Since \( WX=ZY = 2\sqrt{10}\), and \( WZ=XY=\sqrt{10}\) (by distance formula between two points \( (x_1,y_1)\) and \( (x_2,y_2)\): \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)).

Step2: Calculate the perimeter

Perimeter of rectangle \( P=2(WX + WZ)\).
Substitute \( WX = 2\sqrt{10}\) and \( WZ=\sqrt{10}\) into the formula.
\( P=2(2\sqrt{10}+\sqrt{10})=2\times3\sqrt{10}=6\sqrt{10}\) (Wrong again. Wait, no! Wait, let's re - check.
Wait, using the correct distance formula for \( W(-6,6)\) and \( Z(-9,5)\):
\( WZ=\sqrt{(-9+6)^2+(5 - 6)^2}=\sqrt{(-3)^2+(-1)^2}=\sqrt{9 + 1}=\sqrt{10}\).
Since \( WX = ZY=2\sqrt{10}\) (given) and \( WZ=XY=\sqrt{10}\) (by distance formula for two points).
Perimeter \( P=2(WX+WZ)\)
\( P = 2(2\sqrt{10}+\sqrt{10})=2\times3\sqrt{10}=6\sqrt{10}\) (No! Wait, no. Wait, if \( W(-6,6)\), \( X(-4,0)\), \( Y(-7,-1)\), \( Z(-9,5)\)
\( WX=\sqrt{(-4 + 6)^2+(0 - 6)^2}=\sqrt{4+36}=\sqrt{40}=2\sqrt{10}\)
\( XY=\sqrt{(-7 + 4)^2+(-1-0)^2}=\sqrt{9 + 1}=\sqrt{10}\)
Perimeter \( P=2(WX + XY)=2(2\sqrt{10}+\sqrt{10})=6\sqrt{10}\)

Answer:

\(6\sqrt{10}\text{ units}\)