2. Show that the points (-1,3), (-3,-2), and (4,1) are vertices of an isosceles triangle

To show that the points \((-1, 3)\), \((-3, -2)\), and \((4, 1)\) are vertices of an isosceles triangle, we need to confirm that two sides of the triangle have equal length. We can calculate the lengths of the sides using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Let's label the points:
- \(A(-1, 3)\)
- \(B(-3, -2)\)
- \(C(4, 1)\)

### Step 1: Find the distance between points \(A\) and \(B\):
\[
AB = \sqrt{((-3) - (-1))^2 + ((-2) - 3)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}
\]

### Step 2: Find the distance between points \(B\) and \(C\):
\[
BC = \sqrt{(4 - (-3))^2 + (1 - (-2))^2} = \sqrt{(4 + 3)^2 + (1 + 2)^2} = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}
\]

### Step 3: Find the distance between points \(A\) and \(C\):
\[
AC = \sqrt{(4 - (-1))^2 + (1 - 3)^2} = \sqrt{(4 + 1)^2 + (1 - 3)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\]

### Step 4: Compare the distances:
- \(AB = \sqrt{29}\)
- \(BC = \sqrt{58}\)
- \(AC = \sqrt{29}\)

Since \(AB = AC = \sqrt{29}\), two sides of the triangle are equal, which means the triangle is isosceles.

Thus, the points \((-1, 3)\), \((-3, -2)\), and \((4, 1)\) are the vertices of an isosceles triangle.

Would you like further details or have any questions?

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Here are 5 related questions to consider:
1. How would you verify if a triangle is equilateral using the distance formula?
2. Can you prove that the triangle is not a right triangle based on these points?
3. What are the possible coordinates of a point that would make the triangle scalene?
4. How would the formula change if the triangle were located in 3-dimensional space?
5. Can you calculate the perimeter of the triangle using the distances?

**Tip**: When working with the distance formula, always double-check the signs of your coordinates to avoid errors in calculation.

Show that the points (-1, 3), (-3, -2), and (4, 1) are vertices of an isosceles triangle.

Learn how to show that the points (-1, 3), (-3, -2), and (4, 1) form an isosceles triangle by applying the distance formula. Detailed steps include calculating the lengths of the sides and confirming that two sides are equal, as per the Isosceles Triangle Theorem. This solution is suitable for students in Grades 9-10.

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