Question

Use the coordinates to compute the perimeter of the triangle.
Coordinates: (1,1) (3,4) (5,1) A. 10 units B. 11.2 units C. 10.8 units D. 12 units

Answer 1

To compute the perimeter of the triangle formed by the points \( A(1, 1) \), \( B(3, 4) \), and \( C(5, 1) \), we need to find the lengths of the three sides of the triangle. We can do this using the distance formula:

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

### Step 1: Calculate the lengths of the sides.

1. **Length of side \( AB \)** between points \( A(1, 1) \) and \( B(3, 4) \):
\[
AB = \sqrt{(3 - 1)^2 + (4 - 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]

2. **Length of side \( BC \)** between points \( B(3, 4) \) and \( C(5, 1) \):
\[
BC = \sqrt{(5 - 3)^2 + (1 - 4)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

3. **Length of side \( AC \)** between points \( A(1, 1) \) and \( C(5, 1) \):
\[
AC = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4
\]

### Step 2: Calculate the perimeter.

Now, we sum the lengths of the sides to find the perimeter:
\[
\text{Perimeter} = AB + BC + AC = \sqrt{13} + \sqrt{13} + 4 = 2\sqrt{13} + 4
\]

### Step 3: Evaluate \( 2\sqrt{13} + 4 \) numerically.

Calculating \( \sqrt{13} \) approximately:
\[
\sqrt{13} \approx 3.60555
\]
Thus:
\[
2\sqrt{13} \approx 2 \times 3.60555 \approx 7.2111
\]
Adding 4:
\[
\text{Perimeter} \approx 7.2111 + 4 \approx 11.2111
\]

### Conclusion

Rounding \( 11.2111 \) to one decimal place gives about \( 11.2 \) units. Therefore, the perimeter of the triangle is approximately:

\[
\text{Perimeter} \approx 11.2 \text{ units}
\]

The answer is \( \boxed{11.2} \) units (Option B).