Problem 8 - The Calculus with Analytic Geometry

Question

Prove that the points \(A(6,-13), B(-2,2), C(13,10)\), and \(D(21,-5)\) are the vertices of a square. Find the length of a diagonal.

Step-by-Step Solution

Answer

The points form a square, and the length of a diagonal is \(\text{Length of diagonal} = \sqrt{578} \).

Step 1 - Calculate the distance between points

Use the distance formula to find the distances between each pair of points. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 2 - Find distances AB, BC, CD, and DA

Calculate \( AB = \sqrt{(-2 - 6)^2 + (2 - (-13))^2} = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \).Calculate \( BC = \sqrt{(13 - (-2))^2 + (10 - 2)^2} = \sqrt{(15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \).Calculate \( CD = \sqrt{(21 - 13)^2 + (-5 - 10)^2} = \sqrt{(8)^2 + (-15)^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \).Calculate \( DA = \sqrt{(21 - 6)^2 + (-5 - (-13))^2} = \sqrt{(15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \).

Step 3 - Find distances AC and BD (diagonals)

Calculate \( AC = \sqrt{(13 - 6)^2 + (10 - (-13))^2} = \sqrt{(7)^2 + 23^2} = \sqrt{49 + 529} = \sqrt{578} \).Calculate \( BD = \sqrt{(21 - (-2))^2 + (-5 - 2)^2} = \sqrt{(23)^2 + (-7)^2} = \sqrt{529 + 49} = \sqrt{578} \).

Step 4 - Verify the points form a square

To confirm a quadrilateral is a square, all sides must be equal and the diagonals must be equal. As calculated, \(AB = BC = CD = DA = 17 \) and the diagonals \(AC = BD = \sqrt{578} \). Therefore, the points form a square.

Step 5 - Find the length of the diagonal

From the calculations, the length of a diagonal, which is either \(AC \) or \(BD \), is \(\text{Length of diagonal} = \sqrt{578} \).

Key Concepts

Distance formula

To determine if the given points form a square, we first calculate the distances between each pair of points. The distance formula is essential here.
The formula for the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Applying this formula, we find the four side distances and two diagonals:

• \( AB = \sqrt{(6 - (-2))^2 + (-13 - 2)^2} = 17 \)
• \( BC = \sqrt{(13 - (-2))^2 + (10 - 2)^2} = 17 \)
• \( CD = \sqrt{(21 - 13)^2 + (-5 - 10)^2} = 17 \)
• \( DA = \sqrt{(21 - 6)^2 + (-5 - (-13))^2} = 17 \)
• \( AC = \sqrt{(13 - 6)^2 + (10 - (-13))^2} = \sqrt{578} \)
• \( BD = \sqrt{(21 - (-2))^2 + (-5 - 2)^2} = \sqrt{578} \)

These distances help us confirm the shape of the quadrilateral.

Diagonals of a square

The diagonals of a square have important properties. They bisect each other at right angles, are equal in length, and split the square into two congruent right triangles.
Using the points provided:
Calculate the diagonals:

• \( AC = \sqrt{(13 - 6)^2 + (10 - (-13))^2} = \sqrt{578} \)
• \( BD = \sqrt{(21 - (-2))^2 + (-5 - 2)^2} = \sqrt{578} \)

Since the diagonals \( AC \) and \( BD \) are equal in length (both \( \sqrt{578} \)), this is a crucial piece of evidence confirming our points form a square. The equality of the diagonals differentiates squares from other quadrilaterals, like rhombuses, which also have equal sides but do not necessarily have equal diagonals.

Properties of a square

A square has unique properties that distinguish it from other shapes:

• All four sides are equal in length
• All interior angles are right angles (90 degrees)
• The diagonals are equal and bisect each other at right angles

For our points \(A(6,-13), B(-2,2), C(13,10), D(21,-5)\):
The side lengths \( AB, BC, CD, \) and \(DA\) were all calculated to be 17 units.
The diagonals \( AC \) and \( BD \) were both found to be \( \sqrt{578} \).
These characteristics affirm the quadrilateral is indeed a square.
Remember, confirming both equal sides and equal diagonals is necessary to definitively prove a quadrilateral is a square.