Question

finding the length of the sides of a quadrilateral using the distance formula
which statement is true about quadrilateral qrst?
side qr has a length of 5 units.
side rs has a length of \sqrt{26} units.
side st has a length of 6 units.
side tq has a length of \sqrt{17} units.

Explanation:

To solve this, we first identify the coordinates of each point:

• \( Q(-3, 2) \)
• \( R(3, 4) \)
• \( S(3, -2) \)
• \( T(-2, -2) \)

We use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for non - vertical/non - horizontal lines and the formula \( d=\vert x_2 - x_1\vert\) (for horizontal lines, since \( y_2 - y_1 = 0\)) or \( d=\vert y_2 - y_1\vert\) (for vertical lines, since \( x_2 - x_1=0\))

Step 1: Analyze Side QR

For points \( Q(-3,2) \) and \( R(3,4) \)
Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
\( x_1=-3,y_1 = 2,x_2=3,y_2 = 4 \)
\( d_{QR}=\sqrt{(3 - (-3))^2+(4 - 2)^2}=\sqrt{(6)^2+(2)^2}=\sqrt{36 + 4}=\sqrt{40}
eq5\)

Step 2: Analyze Side RS

For points \( R(3,4) \) and \( S(3,-2) \)
Since the \( x\) - coordinates are the same (\( x_1=x_2 = 3\)), this is a vertical line.
The distance formula for vertical lines is \( d=\vert y_2 - y_1\vert\)
\( y_1 = 4,y_2=-2 \)
\( d_{RS}=\vert-2 - 4\vert=\vert-6\vert = 6
eq\sqrt{40}\)

Step 3: Analyze Side ST

For points \( S(3,-2) \) and \( T(-2,-2) \)
Since the \( y\) - coordinates are the same (\( y_1 = y_2=-2\)), this is a horizontal line.
The distance formula for horizontal lines is \( d=\vert x_2 - x_1\vert\)
\( x_1 = 3,x_2=-2 \)
\( d_{ST}=\vert-2 - 3\vert=\vert-5\vert = 5
eq6\)

Step 4: Analyze Side TQ

For points \( T(-2,-2) \) and \( Q(-3,2) \)
Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
\( x_1=-2,y_1=-2,x_2=-3,y_2 = 2 \)
\( d_{TQ}=\sqrt{(-3-(-2))^2+(2 - (-2))^2}=\sqrt{(-1)^2+(4)^2}=\sqrt{1 + 16}=\sqrt{17}\)

So the statement "Side TQ has a length of \(\sqrt{17}\) units" is true.

Answer:

Side TQ has a length of \(\sqrt{17}\) units.