**ANALYZE** Find the perimeter of equilateral triangle *KLM* given the vertices K(−2,1)K-2,1 and M(10,6).M10,6. Explain your reasoning.
 
Perimeter=Perimeter=  units
 
An equilateral triangle has congruent side lengths. Use the Distance Formula to find KM=KM=. So, P=P=(KM)=KM= units.

To find the perimeter of the equilateral triangle \( KLM \), we need to determine the length of one side of the triangle (since all sides are equal) and then multiply that length by 3.

### Step 1: Use the Distance Formula to Find \( KM \)
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the Distance Formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Given points \( K(-2, 1) \) and \( M(10, 6) \), we substitute the coordinates into the Distance Formula:
\[
KM = \sqrt{(10 - (-2))^2 + (6 - 1)^2}
\]
Simplify the expression:
\[
KM = \sqrt{(10 + 2)^2 + (6 - 1)^2} = \sqrt{12^2 + 5^2}
\]
\[
KM = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{units}
\]

### Step 2: Determine the Perimeter
Since \( KLM \) is an equilateral triangle, all sides are equal in length. Thus, the perimeter \( P \) is:
\[
P = 3 \times KM
\]
Substitute the value of \( KM \):
\[
P = 3 \times 13 = 39 \, \text{units}
\]

### Final Answer
The perimeter of the equilateral triangle \( KLM \) is **39 units**.

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Do you need further explanations or have any questions?

Here are 5 related questions you might find interesting:
1. How would you calculate the perimeter if the triangle were not equilateral?
2. Can you determine the length of the third side \( KL \) or \( LM \) using only the coordinates of \( K \) and \( M \)?
3. How would you find the area of the equilateral triangle \( KLM \)?
4. What if the triangle \( KLM \) was located in 3D space? How would you find the perimeter then?
5. Can you find the equation of the circle circumscribing triangle \( KLM \)?

**Tip:** When working with distances in coordinate geometry, always ensure you're correctly substituting and simplifying the Distance Formula to avoid calculation errors.

Learn how to calculate the perimeter of equilateral triangle KLM with detailed steps using the Distance Formula. Understand the properties of equilateral triangles and apply them to find the perimeter. Suitable for Grades 9-12.

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