Question

a grid map marks the plot of harold’s garden in meters. the coordinates of the quadrilateral - shaped property are g(-8, 3), a(4, 8), r(10, 0), and d(-2, -5). he wants to build a short fence around the garden. the perimeter of his garden is ▼ meters.

Explanation:

To find the perimeter of the quadrilateral, we need to calculate the lengths of all four sides (GA, AR, RD, and DG) using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) and then sum them up.

Step 1: Calculate the length of GA

Coordinates of G(-8, 3) and A(4, 8).
\[

$$\begin{align*} GA &= \sqrt{(4 - (-8))^2 + (8 - 3)^2} \\ &= \sqrt{(12)^2 + (5)^2} \\ &= \sqrt{144 + 25} \\ &= \sqrt{169} \\ &= 13 \end{align*}$$

\]

Step 2: Calculate the length of AR

Coordinates of A(4, 8) and R(10, 0).
\[

$$\begin{align*} AR &= \sqrt{(10 - 4)^2 + (0 - 8)^2} \\ &= \sqrt{(6)^2 + (-8)^2} \\ &= \sqrt{36 + 64} \\ &= \sqrt{100} \\ &= 10 \end{align*}$$

\]

Step 3: Calculate the length of RD

Coordinates of R(10, 0) and D(-2, -5).
\[

$$\begin{align*} RD &= \sqrt{(-2 - 10)^2 + (-5 - 0)^2} \\ &= \sqrt{(-12)^2 + (-5)^2} \\ &= \sqrt{144 + 25} \\ &= \sqrt{169} \\ &= 13 \end{align*}$$

\]

Step 4: Calculate the length of DG

Coordinates of D(-2, -5) and G(-8, 3).
\[

$$\begin{align*} DG &= \sqrt{(-8 - (-2))^2 + (3 - (-5))^2} \\ &= \sqrt{(-6)^2 + (8)^2} \\ &= \sqrt{36 + 64} \\ &= \sqrt{100} \\ &= 10 \end{align*}$$

\]

Step 5: Calculate the perimeter

Perimeter = GA + AR + RD + DG
\[

$$\begin{align*} \text{Perimeter} &= 13 + 10 + 13 + 10 \\ &= 46 \end{align*}$$

\]

Answer:

46