Question

A triangular plot of land is shown on the coordinate plane below, where each unit represents 100 feet.

On coordinate plane, A-B-C is a triangle with the points A (4, 5), B (-6, -3) and C (-4, 5).

A surveyor needed to determine the lengths of each side of the land in order to complete the survey.

Complete the table to determine the lengths of each of the sides of the land. Round each answer to the nearest foot.
Side Length (in feet)
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Answer 1

To find the lengths of the sides of the triangular plot of land represented by the points A (4, 5), B (-6, -3), and C (-4, 5) on the coordinate plane, we can use the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

### Step 1: Calculate Length of Side AB
Points:
- A (4, 5)
- B (-6, -3)

Using the distance formula:
\[
AB = \sqrt{((-6) - 4)^2 + ((-3) - 5)^2}
\]
\[
AB = \sqrt{(-10)^2 + (-8)^2}
\]
\[
AB = \sqrt{100 + 64}
\]
\[
AB = \sqrt{164}
\]
\[
AB \approx 12.81
\]
Rounding to the nearest foot:
\[
AB \approx 13 \text{ feet}
\]

### Step 2: Calculate Length of Side BC
Points:
- B (-6, -3)
- C (-4, 5)

Using the distance formula:
\[
BC = \sqrt{((-4) - (-6))^2 + (5 - (-3))^2}
\]
\[
BC = \sqrt{(2)^2 + (8)^2}
\]
\[
BC = \sqrt{4 + 64}
\]
\[
BC = \sqrt{68}
\]
\[
BC \approx 8.25
\]
Rounding to the nearest foot:
\[
BC \approx 8 \text{ feet}
\]

### Step 3: Calculate Length of Side CA
Points:
- C (-4, 5)
- A (4, 5)

Using the distance formula:
\[
CA = \sqrt{((4) - (-4))^2 + (5 - 5)^2}
\]
\[
CA = \sqrt{(8)^2 + (0)^2}
\]
\[
CA = \sqrt{64}
\]
\[
CA = 8
\]
Since this is already an integer, we keep it as:
\[
CA = 8 \text{ feet}
\]

### Summary of Results
Now, we summarize the lengths of the sides in the table:

| Side | Length (in feet) |
|------|------------------|
| AB | 13 |
| BC | 8 |
| CA | 8 |

This table provides the lengths of each side of the triangular plot of land rounded to the nearest foot.