Question

Use the coordinates to compute the perimeter of the triangle. (1 point) Responses 10.8 units 10.8 units 10 units 10 units 11.2 units 11.2 units 12 units the coords are (3,4) (1,1) (5,1)

Answer 1

To compute the perimeter of the triangle formed by the coordinates \((3, 4)\), \((1, 1)\), and \((5, 1)\), we first need to calculate the lengths of each side of the triangle using the distance formula:

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Let's denote the points as follows:
- Point A: \((3, 4)\)
- Point B: \((1, 1)\)
- Point C: \((5, 1)\)

### Calculating Distances

1. **Distance AB:**
\[
AB = \sqrt{(1 - 3)^2 + (1 - 4)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

2. **Distance BC:**
\[
BC = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4
\]

3. **Distance CA:**
\[
CA = \sqrt{(5 - 3)^2 + (1 - 4)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

### Calculating the Perimeter
Now, we can calculate the perimeter \(P\) of the triangle by summing the lengths of the sides:

\[
P = AB + BC + CA = \sqrt{13} + 4 + \sqrt{13} = 2\sqrt{13} + 4
\]

Now, let's approximate \(\sqrt{13}\):
\[
\sqrt{13} \approx 3.605
\]
So,
\[
P \approx 2(3.605) + 4 = 7.21 + 4 \approx 11.21
\]

Thus, the perimeter of the triangle is approximately **11.2 units**.

Therefore, the correct response is:
**11.2 units**.