ΔABC is located at A (3, 1), B (4, 4), and C (5, 1). Zackery says that ΔABC is an equilateral triangle, while Verna says that it is an isosceles triangle. Who is correct? (1 point)

To determine whether the triangle \(\Delta ABC\) is equilateral or isosceles, we will calculate the lengths of the sides of the triangle. An equilateral triangle has all sides of equal length, while an isosceles triangle has at least two sides of equal length.

### Step 1: Calculate the length of each side of \(\Delta ABC\)

We will use the distance formula to find the lengths of the sides of the triangle. The distance formula is:

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

#### Side \(AB\) (between points \(A(3, 1)\) and \(B(4, 4)\)):
\[
AB = \sqrt{(4 - 3)^2 + (4 - 1)^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]

#### Side \(BC\) (between points \(B(4, 4)\) and \(C(5, 1)\)):
\[
BC = \sqrt{(5 - 4)^2 + (1 - 4)^2} = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

#### Side \(AC\) (between points \(A(3, 1)\) and \(C(5, 1)\)):
\[
AC = \sqrt{(5 - 3)^2 + (1 - 1)^2} = \sqrt{2^2 + 0^2} = \sqrt{4} = 2
\]

### Step 2: Analyze the results

- \(AB = \sqrt{10}\)
- \(BC = \sqrt{10}\)
- \(AC = 2\)

Since \(AB = BC\), the triangle has two equal sides, meaning it is **isosceles**.

### Conclusion:
Verna is correct. \(\Delta ABC\) is an **isosceles triangle**.

Would you like additional explanation or a similar problem to work through? Here are 5 related questions to help you explore further:

1. What would happen if all three sides of the triangle were equal in length?
2. How do you determine if a triangle is scalene?
3. Can an equilateral triangle also be classified as isosceles?
4. How do you find the area of a triangle using its vertices?
5. What other properties can help identify different types of triangles besides the side lengths?

**Tip:** When analyzing triangles, always start by calculating the lengths of the sides to classify the type of triangle accurately.

ΔABC is located at A (3, 1), B (4, 4), and C (5, 1). Zackery says that ΔABC is an equilateral triangle, while Verna says that it is an isosceles triangle. Who is correct?

Solve the problem of classifying triangle ΔABC with vertices A(3,1), B(4,4), and C(5,1). Use the distance formula to determine whether the triangle is equilateral or isosceles. This detailed solution covers key concepts in geometry and triangle classification, ideal for students in Grades 8-10.

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