Question

1. find the perimeter of pentagon j k l m n round to the nearest tenth if necessary. j(-1, -5), k(-4, 2), l(0, 2), m(3, 6), n(5, -5)

Explanation:

To find the perimeter of pentagon \(JKLMN\), we need to calculate the length of each side using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\): \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), and then sum all the side lengths.

Step 1: Calculate \(JK\)

Points \(J(-1, -5)\) and \(K(-4, 2)\)
\[

$$\begin{align*} JK &= \sqrt{(-4 - (-1))^2 + (2 - (-5))^2}\\ &= \sqrt{(-3)^2 + (7)^2}\\ &= \sqrt{9 + 49}\\ &= \sqrt{58}\\ &\approx 7.6 \end{align*}$$

\]

Step 2: Calculate \(KL\)

Points \(K(-4, 2)\) and \(L(0, 2)\)
\[

$$\begin{align*} KL &= \sqrt{(0 - (-4))^2 + (2 - 2)^2}\\ &= \sqrt{(4)^2 + (0)^2}\\ &= \sqrt{16 + 0}\\ &= \sqrt{16}\\ &= 4 \end{align*}$$

\]

Step 3: Calculate \(LM\)

Points \(L(0, 2)\) and \(M(3, 6)\)
\[

$$\begin{align*} LM &= \sqrt{(3 - 0)^2 + (6 - 2)^2}\\ &= \sqrt{(3)^2 + (4)^2}\\ &= \sqrt{9 + 16}\\ &= \sqrt{25}\\ &= 5 \end{align*}$$

\]

Step 4: Calculate \(MN\)

Points \(M(3, 6)\) and \(N(5, -5)\)
\[

$$\begin{align*} MN &= \sqrt{(5 - 3)^2 + (-5 - 6)^2}\\ &= \sqrt{(2)^2 + (-11)^2}\\ &= \sqrt{4 + 121}\\ &= \sqrt{125}\\ &\approx 11.2 \end{align*}$$

\]

Step 5: Calculate \(NJ\)

Points \(N(5, -5)\) and \(J(-1, -5)\)
\[

$$\begin{align*} NJ &= \sqrt{(-1 - 5)^2 + (-5 - (-5))^2}\\ &= \sqrt{(-6)^2 + (0)^2}\\ &= \sqrt{36 + 0}\\ &= \sqrt{36}\\ &= 6 \end{align*}$$

\]

Step 6: Sum all side lengths

\[

$$\begin{align*} \text{Perimeter} &= JK + KL + LM + MN + NJ\\ &\approx 7.6 + 4 + 5 + 11.2 + 6\\ &= 33.8 \end{align*}$$

\]

Answer:

The perimeter of pentagon \(JKLMN\) is \(\boxed{33.8}\).