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.