Are the points (−2, 10), (4, 7), (1, 1), and (−5, 4) the vertices of a square?

To determine whether the points \((-2, 10)\), \((4, 7)\), \((1, 1)\), and \((-5, 4)\) are the vertices of a square, we need to verify two main conditions:

1. **All sides of the quadrilateral formed by these points are equal in length**.
2. **The diagonals of the quadrilateral are equal in length**.

### 1. Calculate the Lengths of the Sides

We'll use the distance formula to calculate the length between each pair of points:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Calculate all four sides:
- Between \((-2, 10)\) and \((4, 7)\):
  \[
  \text{Side 1} = \sqrt{(4 - (-2))^2 + (7 - 10)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
  \]

- Between \((4, 7)\) and \((1, 1)\):
  \[
  \text{Side 2} = \sqrt{(1 - 4)^2 + (1 - 7)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}
  \]

- Between \((1, 1)\) and \((-5, 4)\):
  \[
  \text{Side 3} = \sqrt{(-5 - 1)^2 + (4 - 1)^2} = \sqrt{(-6)^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
  \]

- Between \((-5, 4)\) and \((-2, 10)\):
  \[
  \text{Side 4} = \sqrt{(-2 - (-5))^2 + (10 - 4)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}
  \]

Since all four sides are equal (\(3\sqrt{5}\)), the quadrilateral has equal sides, which is a necessary condition for it to be a square.

### 2. Calculate the Lengths of the Diagonals

Now, calculate the lengths of the diagonals:

- Diagonal between \((-2, 10)\) and \((1, 1)\):
  \[
  \text{Diagonal 1} = \sqrt{(1 - (-2))^2 + (1 - 10)^2} = \sqrt{3^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
  \]

- Diagonal between \((4, 7)\) and \((-5, 4)\):
  \[
  \text{Diagonal 2} = \sqrt{(-5 - 4)^2 + (4 - 7)^2} = \sqrt{(-9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}
  \]

Since both diagonals are equal (\(3\sqrt{10}\)), the quadrilateral also satisfies the diagonal condition for a square.

### Conclusion

Because the sides are all equal and the diagonals are equal, the points \((-2, 10)\), \((4, 7)\), \((1, 1)\), and \((-5, 4)\) **do indeed form a square**.

Do you have any questions about this conclusion, or would you like more details?

Here are some related questions:
1. How can you verify if four points form a rectangle?
2. What are the conditions for four points to form a rhombus?
3. How do you find the area of a square given its vertices?
4. What other shapes can be verified using distance formulas?
5. Can the distance formula be used to determine if four points form a parallelogram?

**Tip:** When checking if a quadrilateral is a square, equal sides and equal diagonals are both necessary conditions.

Explore whether the points (-2, 10), (4, 7), (1, 1), and (-5, 4) form a square using mathematical analysis. Check the equality of sides and diagonals using the distance formula. Suitable for Grades 9-12.

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