A triangle has vertices at S(6, 3), T(−1, 10), and R(−4, −5). What is the perimeter of the
triangle? Sketch the triangle on graph paper and show the work that leads to your answer –
leave your answer in simplest radical form.

To find the perimeter of the triangle with vertices \( S(6, 3) \), \( T(-1, 10) \), and \( R(-4, -5) \), we need to calculate the lengths of the sides \( ST \), \( TR \), and \( RS \).

First, we will use the distance formula to find the lengths of the sides:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Calculate the length of \( ST \)
\[ S(6, 3) \quad T(-1, 10) \]
\[ ST = \sqrt{(-1 - 6)^2 + (10 - 3)^2} \]
\[ ST = \sqrt{(-7)^2 + (7)^2} \]
\[ ST = \sqrt{49 + 49} \]
\[ ST = \sqrt{98} \]
\[ ST = 7\sqrt{2} \]

### Step 2: Calculate the length of \( TR \)
\[ T(-1, 10) \quad R(-4, -5) \]
\[ TR = \sqrt{(-4 - (-1))^2 + (-5 - 10)^2} \]
\[ TR = \sqrt{(-4 + 1)^2 + (-5 - 10)^2} \]
\[ TR = \sqrt{(-3)^2 + (-15)^2} \]
\[ TR = \sqrt{9 + 225} \]
\[ TR = \sqrt{234} \]
\[ TR = 3\sqrt{26} \]

### Step 3: Calculate the length of \( RS \)
\[ R(-4, -5) \quad S(6, 3) \]
\[ RS = \sqrt{(6 - (-4))^2 + (3 - (-5))^2} \]
\[ RS = \sqrt{(6 + 4)^2 + (3 + 5)^2} \]
\[ RS = \sqrt{10^2 + 8^2} \]
\[ RS = \sqrt{100 + 64} \]
\[ RS = \sqrt{164} \]
\[ RS = 2\sqrt{41} \]

### Step 4: Calculate the perimeter
The perimeter \( P \) is the sum of the lengths of the sides:
\[ P = ST + TR + RS \]
\[ P = 7\sqrt{2} + 3\sqrt{26} + 2\sqrt{41} \]

This is the perimeter of the triangle in simplest radical form.

### Sketch the Triangle
To sketch the triangle, plot the points \( S(6, 3) \), \( T(-1, 10) \), and \( R(-4, -5) \) on graph paper and draw the lines connecting these points.

Would you like any further details or have any questions about the solution?

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Here are 8 related questions for further practice:

1. How do you find the midpoint of a line segment given two endpoints?
2. What is the formula for the area of a triangle given its vertices?
3. How do you determine if three points are collinear?
4. Can you find the equation of the line passing through points \( S \) and \( T \)?
5. What is the distance between the point \( (3, 4) \) and the line \( y = 2x + 1 \)?
6. How do you find the centroid of a triangle given its vertices?
7. What is the slope of the line segment connecting \( S \) and \( R \)?
8. How do you find the circumcenter of a triangle given its vertices?

**Tip:** Always double-check your calculations to ensure accuracy, especially when working with square roots and radicals.

Learn how to calculate the perimeter of a triangle with vertices at S(6, 3), T(-1, 10), and R(-4, -5) using the distance formula. This problem involves finding the lengths of the sides and summing them to find the perimeter in simplest radical form. Suitable for students in Grades 9-12.

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