Question

Use distance formula to find the base and height of this rectangle, then calculate its perimeter (A -3,-1 B -2,-4 C 4,-2 D 3,1). Responses P = 50 units P = 50 units P = 610−−√ or about 19.0 units P = 6 square root of 10 or about 19.0 units P = 100 units P = 100 units P = 310−−√ or about 9.5 units

Answer 1

To calculate the perimeter of the rectangle formed by the points A(-3, -1), B(-2, -4), C(4, -2), and D(3, 1), we first need to find the lengths of two adjacent sides of the rectangle (the base and the height).

### Step 1: Calculate the lengths of the sides
We will use the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Finding the length of AB:
- Point A: (-3, -1)
- Point B: (-2, -4)

Using the distance formula:
\[
AB = \sqrt{((-2) - (-3))^2 + ((-4) - (-1))^2} = \sqrt{(-2 + 3)^2 + (-4 + 1)^2}
\]
\[
= \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \text{ units}
\]

#### Finding the length of AD:
- Point A: (-3, -1)
- Point D: (3, 1)

Using the distance formula:
\[
AD = \sqrt{(3 - (-3))^2 + (1 - (-1))^2} = \sqrt{(3 + 3)^2 + (1 + 1)^2}
\]
\[
= \sqrt{(6)^2 + (2)^2} = \sqrt{36 + 4} = \sqrt{40} \text{ units} = \sqrt{4 \times 10} = 2\sqrt{10} \text{ units}
\]

### Step 2: Calculate the perimeter
The perimeter \(P\) of a rectangle is given by:
\[
P = 2 \times (\text{base} + \text{height})
\]
Taking the lengths we calculated:
- Base (AB) = \(\sqrt{10}\) units
- Height (AD) = \(2\sqrt{10}\) units

Thus, the perimeter is:
\[
P = 2 \times (\sqrt{10} + 2\sqrt{10}) = 2 \times 3\sqrt{10} = 6\sqrt{10} \text{ units}
\]

### Conclusion
The calculated perimeter is:
\[
P = 6\sqrt{10} \text{ units}
\]

This corresponds to the option \(P = 6\sqrt{10}\) or about 19.0 units.