Question

lets try: find the perimeter of qrst. round to the nearest tenth if necessary. q(6, 0), r(-3, -5), s(-1, 4), t(2, 0)

Explanation:

Step1: Recall the distance formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). We need to find the lengths of \(QR\), \(RS\), \(ST\), and \(TQ\) and sum them up to get the perimeter.

Step2: Calculate \(QR\) (Q(6,0) to R(-3,-5))

\[

$$\begin{align*} QR&=\sqrt{(-3 - 6)^2 + (-5 - 0)^2}\\ &=\sqrt{(-9)^2 + (-5)^2}\\ &=\sqrt{81 + 25}\\ &=\sqrt{106}\\ &\approx 10.3 \end{align*}$$

\]

Step3: Calculate \(RS\) (R(-3,-5) to S(-1,4))

\[

$$\begin{align*} RS&=\sqrt{(-1 - (-3))^2 + (4 - (-5))^2}\\ &=\sqrt{(2)^2 + (9)^2}\\ &=\sqrt{4 + 81}\\ &=\sqrt{85}\\ &\approx 9.2 \end{align*}$$

\]

Step4: Calculate \(ST\) (S(-1,4) to T(2,0))

\[

$$\begin{align*} ST&=\sqrt{(2 - (-1))^2 + (0 - 4)^2}\\ &=\sqrt{(3)^2 + (-4)^2}\\ &=\sqrt{9 + 16}\\ &=\sqrt{25}\\ &= 5 \end{align*}$$

\]

Step5: Calculate \(TQ\) (T(2,0) to Q(6,0))

Since the \(y\)-coordinates are the same, the distance is the absolute difference of \(x\)-coordinates:
\[
TQ = |6 - 2| = 4
\]

Step6: Sum the lengths to find the perimeter

Perimeter \(= QR + RS + ST + TQ\)
\[

$$\begin{align*} &\approx 10.3 + 9.2 + 5 + 4\\ &= 28.5 \end{align*}$$

\]

Answer:

The perimeter of \(QRST\) is approximately \(28.5\).