Question

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

Answer 1

To compute the perimeter of the triangle with the given vertices at coordinates (1,6), (1,2), and (-4,2), we first need to find the lengths of each side 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. **Length of side between (1,6) and (1,2)**:
\[
d_1 = \sqrt{(1 - 1)^2 + (2 - 6)^2} = \sqrt{0 + (-4)^2} = \sqrt{16} = 4
\]

2. **Length of side between (1,2) and (-4,2)**:
\[
d_2 = \sqrt{(-4 - 1)^2 + (2 - 2)^2} = \sqrt{(-5)^2 + 0} = \sqrt{25} = 5
\]

3. **Length of side between (-4,2) and (1,6)**:
\[
d_3 = \sqrt{(1 - (-4))^2 + (6 - 2)^2} = \sqrt{(1 + 4)^2 + (4)^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41}
\]

### Step 2: Calculate the perimeter
Now we sum the lengths of the sides to find the perimeter \(P\):

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

Calculating \(\sqrt{41}\):
\[
\sqrt{41} \approx 6.4
\]

Thus, the perimeter becomes:
\[
P \approx 4 + 5 + 6.4 = 15.4
\]

### Step 3: Round to the nearest tenth
The perimeter of the triangle, rounded to the nearest tenth, is:
\[
\boxed{15.4}
\]

Answer 2

now please do the same with this polygon. the coordinates are, (2, 0), (0, 2), (-2, 0), (1, -2), and (-1, -2)

Answer 3

To compute the perimeter of the polygon with the given vertices at coordinates (2, 0), (0, 2), (-2, 0), (1, -2), and (-1, -2), we will calculate the lengths of each side using the distance formula.

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

1. **Length between (2, 0) and (0, 2)**:
\[
d_1 = \sqrt{(0 - 2)^2 + (2 - 0)^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828
\]

2. **Length between (0, 2) and (-2, 0)**:
\[
d_2 = \sqrt{(-2 - 0)^2 + (0 - 2)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828
\]

3. **Length between (-2, 0) and (1, -2)**:
\[
d_3 = \sqrt{(1 - (-2))^2 + (-2 - 0)^2} = \sqrt{(1 + 2)^2 + (-2)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.605
\]

4. **Length between (1, -2) and (-1, -2)**:
\[
d_4 = \sqrt{(-1 - 1)^2 + (-2 - (-2))^2} = \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2
\]

5. **Length between (-1, -2) and (2, 0)**:
\[
d_5 = \sqrt{(2 - (-1))^2 + (0 - (-2))^2} = \sqrt{(2 + 1)^2 + (2)^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.605
\]

### Step 2: Calculate the perimeter
Now we sum the lengths of all the sides to find the perimeter \(P\):

\[
P = d_1 + d_2 + d_3 + d_4 + d_5
\]

Substituting the approximate values:

\[
P \approx 2.828 + 2.828 + 3.605 + 2 + 3.605
\]

Calculating the total:

\[
P \approx 2.828 + 2.828 + 3.605 + 2 + 3.605 \approx 14.876
\]

### Step 3: Round to the nearest tenth
The perimeter of the polygon, rounded to the nearest tenth, is:

\[
\boxed{14.9}
\]