Problem 41 - Algebra and Trigonometry

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 form a square, as all sides are equal \( \sqrt{45} \) and diagonals are equal \( \sqrt{90} \).

Step 1: Calculate Distances

First, we need to calculate the distances between each pair of consecutive points and ensure all sides and diagonals match those of a square. Use the distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Calculate:- \( AB = \sqrt{(4 + 2)^2 + (6 - 9)^2} = \sqrt{36 + 9} = \sqrt{45} \)- \( BC = \sqrt{(1 - 4)^2 + (0 - 6)^2} = \sqrt{9 + 36} = \sqrt{45} \)- \( CD = \sqrt{(1 + 5)^2 + (0 - 3)^2} = \sqrt{36 + 9} = \sqrt{45} \)- \( DA = \sqrt{(-5 + 2)^2 + (3 - 9)^2} = \sqrt{9 + 36} = \sqrt{45} \)

Step 2: Check Diagonals

For a square, the diagonals should also be of equal length.Calculate:- \( AC = \sqrt{(1 + 2)^2 + (0 - 9)^2} = \sqrt{9 + 81} = \sqrt{90} \)- \( BD = \sqrt{(4 + 5)^2 + (6 - 3)^2} = \sqrt{81 + 9} = \sqrt{90} \)Since both diagonals are equal, \( \sqrt{90} \), this criteria for a square is satisfied.

Step 3: Confirm Properties of Square

All sides are equal with \( AB = BC = CD = DA = \sqrt{45} \) and both diagonals are equal with \( AC = BD = \sqrt{90} \). These calculations confirm that the figure is a square.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in geometry used to calculate the distance between two points in a plane. It is applicable in various problems, including determining whether a given set of points form a specific shape, like a square. The formula is derived from the Pythagorean theorem and is written as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here,

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

To apply the distance formula effectively,

• substitute the given coordinates into the formula,
• calculate the differences \((x_2 - x_1)\) and \((y_2 - y_1)\),
• square both differences, add them together, and then take the square root of the result.

This formula provides an accurate way to determine how far apart two points are on the Cartesian plane, crucial for verifying the shape formed by connecting points.

Properties of a Square

Understanding the properties of a square is essential in geometry. A square is a special type of rectangle and rhombus that is characterized by its four equal sides and four right angles. Here are some key properties:

• All sides of a square are of equal length. This means if you calculate the distance between successive vertices (corners) of a square, they should all be identical.
• The internal angles of a square are all 90 degrees, forming perfect right angles.
• A square has two sets of parallel sides.
• A square can be considered both a rectangle, with additional equilateral properties, and a rhombus, with orthogonal angles.
• The lengths of the diagonals are equal and they bisect each other at right angles.

These consistent properties help to easily identify a square and distinguish it from other quadrilaterals.

Equal Sides and Diagonals

In the context of a square, one significant characteristic is having equal sides and diagonals. Once distances between points are calculated using the distance formula, we can confirm if they form the sides of a square provided all are equal. For our example, this is verified as:

• All side lengths, \( AB \), \( BC \), \( CD \), and \( DA \), were found to be \(\sqrt{45}\), showing equality in the sides.
• Similarly, the diagonals \( AC \) and \( BD \) are both \( \sqrt{90} \).
• This equal length of diagonals not only confirms the figure is a quadrilateral with equal sides, but also verifies the bisecting property unique to squares.

This property of having equal lengths of sides and diagonals is unique to squares and is crucial for verification when analyzing geometric shapes in any planar analysis. With equal diagonals, we also ensure symmetry within the shape.