Question

Use the coordinates (-4,2) (1,2) and (1,6) to compute the perimeter of the triangle. Round your answer to the nearest tenth

Answer 1

To find the perimeter of the triangle formed by the points \((-4, 2)\), \( (1, 2)\), and \( (1, 6)\), we will first compute the lengths of the sides of the triangle using the distance formula. The distance \(d\) 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. Side between \((-4, 2)\) and \( (1, 2)\):**

\[
d_1 = \sqrt{(1 - (-4))^2 + (2 - 2)^2}
\]
\[
d_1 = \sqrt{(1 + 4)^2 + 0^2}
\]
\[
d_1 = \sqrt{5^2} = 5
\]

**2. Side between \( (1, 2)\) and \( (1, 6)\):**

\[
d_2 = \sqrt{(1 - 1)^2 + (6 - 2)^2}
\]
\[
d_2 = \sqrt{0^2 + (6 - 2)^2}
\]
\[
d_2 = \sqrt{4^2} = 4
\]

**3. Side between \( (-4, 2)\) and \( (1, 6)\):**

\[
d_3 = \sqrt{(1 - (-4))^2 + (6 - 2)^2}
\]
\[
d_3 = \sqrt{(1 + 4)^2 + (6 - 2)^2}
\]
\[
d_3 = \sqrt{5^2 + 4^2}
\]
\[
d_3 = \sqrt{25 + 16} = \sqrt{41}
\]

### Step 2: Compute the perimeter.

Now we can find the perimeter \(P\) of the triangle by adding the lengths of the sides:

\[
P = d_1 + d_2 + d_3 = 5 + 4 + \sqrt{41}
\]

We can approximate \(\sqrt{41}\):

\[
\sqrt{41} \approx 6.4 \quad (\text{since } 6.4^2 = 40.96 \text{ is close to } 41)
\]

Therefore, substituting this back into the perimeter equation:

\[
P \approx 5 + 4 + 6.4 = 15.4
\]

### Step 3: Round to the nearest tenth.

The perimeter of the triangle is approximately:

\[
\boxed{15.4}
\]