Math Problem Statement

Given: , , , and are the vertices of quadrilateral .

Prove: is a square.

Using the distance formula, I found that .

Solution

Let's proceed with proving that quadrilateral ABCDABCD is a square, using the distance formula. To do this, we will show that:

  1. All sides of quadrilateral ABCDABCD are equal.
  2. The diagonals of quadrilateral ABCDABCD are equal and perpendicular to each other.

Let the vertices of the quadrilateral be A(x1,y1)A(x_1, y_1), B(x2,y2)B(x_2, y_2), C(x3,y3)C(x_3, y_3), and D(x4,y4)D(x_4, y_4).

Step 1: Using the Distance Formula

The distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

We will now calculate the distances between each pair of consecutive vertices:

  1. ABAB d(AB)=(x2x1)2+(y2y1)2d(AB) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  2. BCBC d(BC)=(x3x2)2+(y3y2)2d(BC) = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
  3. CDCD d(CD)=(x4x3)2+(y4y3)2d(CD) = \sqrt{(x_4 - x_3)^2 + (y_4 - y_3)^2}
  4. DADA d(DA)=(x1x4)2+(y1y4)2d(DA) = \sqrt{(x_1 - x_4)^2 + (y_1 - y_4)^2}

For ABCDABCD to be a square, all four sides must be equal, so: d(AB)=d(BC)=d(CD)=d(DA)d(AB) = d(BC) = d(CD) = d(DA)

Step 2: Checking the Diagonals

We also need to check the diagonals ACAC and BDBD for equality and perpendicularity.

  1. ACAC d(AC)=(x3x1)2+(y3y1)2d(AC) = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}
  2. BDBD d(BD)=(x4x2)2+(y4y2)2d(BD) = \sqrt{(x_4 - x_2)^2 + (y_4 - y_2)^2}

For ABCDABCD to be a square, the diagonals must be equal: d(AC)=d(BD)d(AC) = d(BD)

Additionally, for the diagonals to be perpendicular, their slopes must satisfy the condition: slope of AC×slope of BD=1\text{slope of } AC \times \text{slope of } BD = -1 The slope of a line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: slope=y2y1x2x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}

Thus, for ACAC and BDBD: (y3y1x3x1)×(y4y2x4x2)=1\left(\frac{y_3 - y_1}{x_3 - x_1}\right) \times \left(\frac{y_4 - y_2}{x_4 - x_2}\right) = -1

Conclusion:

By showing that:

  1. All four sides of ABCDABCD are equal.
  2. The diagonals ACAC and BDBD are equal and perpendicular.

We can conclude that ABCDABCD is a square.


Would you like me to walk through an example with specific coordinates? Or would you prefer more details on any part of the process?

Here are some related questions:

  1. How do you calculate the midpoint of a diagonal in a quadrilateral?
  2. What is the significance of perpendicular diagonals in quadrilaterals?
  3. Can a quadrilateral be a square if only the sides are equal but the diagonals are not?
  4. What are the properties of a rhombus, and how is it different from a square?
  5. How can you prove that a rectangle is not a square using slopes?

Tip: The perpendicularity of diagonals is a key indicator when proving a square, as it guarantees 90-degree angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Properties of Quadrilaterals

Formulas

Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Slope formula: slope = (y2 - y1) / (x2 - x1)

Theorems

Properties of squares: all sides equal, diagonals equal and perpendicular

Suitable Grade Level

Grades 9-12