Problem 41 - Precalculus Mathematics for Calculus

Question

Show that the points \(A(-2,9), B(4,6), C(1,0),\) and \(D(-5,3)\) are the vertices of a square.

Step-by-Step Solution

Answer

The points are vertices of a square because all sides are equal and perpendicular.

Step 1: Calculate the Distances between Points

To determine if the points form a square, start by calculating the distances between each pair of consecutive vertices: For points \(A(-2, 9)\) and \(B(4, 6)\): \[ AB = \sqrt{(4 - (-2))^2 + (6 - 9)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{45} = 3\sqrt{5} \] For points \(B(4, 6)\) and \(C(1, 0)\): \[ BC = \sqrt{(1 - 4)^2 + (0 - 6)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{45} = 3\sqrt{5} \] For points \(C(1, 0)\) and \(D(-5, 3)\): \[ CD = \sqrt{(-5 - 1)^2 + (3 - 0)^2} = \sqrt{(-6)^2 + 3^2} = \sqrt{45} = 3\sqrt{5} \] For points \(D(-5, 3)\) and \(A(-2, 9)\): \[ DA = \sqrt{((-2) - (-5))^2 + (9 - 3)^2} = \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5} \]

Step 2: Check Diagonal Lengths

Next, calculate the diagonals to ensure that they have equal lengths and form the correct relationship for a square (i.e., diagonals are equal and \(d = \sqrt{2}l\) where \(l\) is the side length):For the diagonal \(AC\): \[ AC = \sqrt{(1 - (-2))^2 + (0 - 9)^2} = \sqrt{3^2 + (-9)^2} = \sqrt{90} = 3\sqrt{10} \] For the diagonal \(BD\): \[ BD = \sqrt{(4 - (-5))^2 + (6 - 3)^2} = \sqrt{9^2 + 3^2} = \sqrt{90} = 3\sqrt{10} \] Both diagonals are equal, confirming part of the condition for a square.

Step 3: Verify Perpendicularity

Finally, confirm that the adjacent sides are perpendicular by checking the slopes of the sides:For \(AB\): Slope \(m_1 = \frac{6 - 9}{4 - (-2)} = \frac{-3}{6} = -\frac{1}{2}\) For \(BC\): Slope \(m_2 = \frac{0 - 6}{1 - 4} = \frac{-6}{-3} = 2\)The slopes \(m_1 \times m_2 = -\frac{1}{2} \times 2 = -1\), indicating perpendicular lines.Similarly, check \(CD\) and \(DA\) to confirm perpendicularity between all sides.

Step 4: Conclusion

Since all sides are equal, the diagonals are equal, and all adjacent sides are perpendicular, the points \(A, B, C,\) and \(D\) form a square.

Key Concepts

Distance Formula

The distance formula helps find the space between two points in a plane, using their coordinates. Knowing this formula is key in analytical geometry, as it helps in determining the relationships between points, such as the sides of a shape. The formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where

• \(d\) is the distance between the points
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points

In the exercise, this formula was used to calculate each side of the quadrilateral formed by points A, B, C, and D. Since all sides have the same length, one condition for identifying a square is met.

Perpendicularity

In geometry, checking perpendicularity involves determining if two lines intersect at a right angle. For lines to be perpendicular, the product of their slopes must equal -1.
In our solution, the slopes are calculated using:

• For line between points \( (x_1, y_1) \) and \( (x_2, y_2) \), slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

By finding the slopes between each pair of consecutive points (e.g., A and B, B and C), and multiplying them, we confirm they equal -1.
This step verified all corner angles of the polygon are right angles, another criterion being met for the quadrilateral to qualify as a square.

Square Properties

A square is a special type of quadrilateral that has the following properties:

• All four sides are of equal length.
• Each angle measures 90 degrees, making all adjacent sides perpendicular.
• Both diagonals of the square are equal in length.

To confirm that the given points form a square, properties of a square were checked:
1. **Equal Sides:** Using the distance formula, the sides were found equal, meeting the first property.
2. **Right Angles:** Perpendicularity of adjacent sides confirmed 90-degree angles.
3. **Equal Diagonals:** Calculating diagonal distances, their equality was verified, reinforcing the identity of the shape as a square.
These confirmations satisfy all conditions necessary to categorize the shape formed by points A, B, C, and D as a square.