Question

Use the given coordinates to compute the perimeter of the triangle. Round your answer to the nearest tenth. R/(-4,2) P-(1,2) Q-(1,6)

Answer 1

To compute the perimeter of the triangle formed by the points \( R(-4, 2) \), \( P(1, 2) \), and \( Q(1, 6) \), we need to find the lengths of the sides 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:

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

### Step 1: Calculate the lengths of each side

1. **Length of \( RP \)**:
- \( R(-4, 2) \) and \( P(1, 2) \)
\[
RP = \sqrt{(1 - (-4))^2 + (2 - 2)^2} = \sqrt{(1 + 4)^2 + 0^2} = \sqrt{5^2} = 5
\]

2. **Length of \( PQ \)**:
- \( P(1, 2) \) and \( Q(1, 6) \)
\[
PQ = \sqrt{(1 - 1)^2 + (6 - 2)^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4
\]

3. **Length of \( QR \)**:
- \( Q(1, 6) \) and \( R(-4, 2) \)
\[
QR = \sqrt{(-4 - 1)^2 + (2 - 6)^2} = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41}
\]

### Step 2: Compute the perimeter

Now we can find the perimeter of the triangle by summing up the lengths of its sides:

\[
\text{Perimeter} = RP + PQ + QR = 5 + 4 + \sqrt{41}
\]

Calculating \(\sqrt{41}\):

\[
\sqrt{41} \approx 6.4 \quad (\text{rounded to the nearest tenth})
\]

Thus,

\[
\text{Perimeter} \approx 5 + 4 + 6.4 = 15.4
\]

### Final result

The perimeter of the triangle, rounded to the nearest tenth, is

\[
\boxed{15.4}
\]