The points Oleft parenthesis, 1, comma, minus, 7, right parenthesis(1,−7), Pleft parenthesis, 1, comma, minus, 1, right parenthesis(1,−1), and Qleft parenthesis, minus, 6, comma, 2, right parenthesis(−6,2) form a triangle. Plot the points then click the "Graph Triangle" button. Then find the perimeter of the triangle. Round your answer to the nearest tenth if necessary.

To solve the problem of finding the perimeter of the triangle formed by the points \(O(1, -7)\), \(P(1, -1)\), and \(Q(-6, 2)\), we'll go through the following steps:

1. **Plot the Points:** To visualize the triangle, we plot the points on a coordinate plane.
   
2. **Calculate the Lengths of the Sides:** We use the distance formula to find the lengths of the sides \(OP\), \(PQ\), and \(QO\).

3. **Calculate the Perimeter:** The perimeter of the triangle is the sum of the lengths of its sides.

### Step 1: Plot the Points
Let's first identify the coordinates:
- \(O(1, -7)\)
- \(P(1, -1)\)
- \(Q(-6, 2)\)

### Step 2: Calculate the Lengths of the Sides
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- **Length \(OP\):**
\[
OP = \sqrt{(1 - 1)^2 + (-1 - (-7))^2} = \sqrt{0^2 + 6^2} = \sqrt{36} = 6
\]

- **Length \(PQ\):**
\[
PQ = \sqrt{(-6 - 1)^2 + (2 - (-1))^2} = \sqrt{(-7)^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58} \approx 7.6
\]

- **Length \(QO\):**
\[
QO = \sqrt{(-6 - 1)^2 + (2 - (-7))^2} = \sqrt{(-7)^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4
\]

### Step 3: Calculate the Perimeter
The perimeter \(P\) of the triangle is the sum of the side lengths:
\[
P = OP + PQ + QO = 6 + 7.6 + 11.4 = 25 \text{ units}
\]

### Final Answer
The perimeter of the triangle is \(25\) units.

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How do you find the area of the triangle using the given points?
2. What would the perimeter be if the coordinates of one point were different?
3. How can you determine if the given points form a right triangle?
4. Can the triangle formed by these points be an isosceles triangle?
5. How do the distances between points change if we translate the entire triangle by a vector?

**Tip:** Always double-check your calculations when using the distance formula, especially with negative coordinates!

Learn how to calculate the perimeter of a triangle formed by points (1, -7), (1, -1), and (-6, 2) using the distance formula. Follow a detailed step-by-step solution to find the lengths of the sides OP, PQ, and QO, and then compute the perimeter. Suitable for high school students.

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