Problem 47 - 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 since all sides are equal and diagonals are equal.

Step 1: Calculate AB, BC, CD, and DA

First, calculate the length of each side of the quadrilateral. Use the distance formula which is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).For \(AB\), \((-2,9)\) and \((4,6)\):\[ AB = \sqrt{(4 - (-2))^2 + (6 - 9)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}\]For \(BC\), \((4,6)\) and \((1,0)\):\[ BC = \sqrt{(1 - 4)^2 + (0 - 6)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\]For \(CD\), \((1,0)\) and \((-5,3)\):\[ CD = \sqrt{(-5 - 1)^2 + (3 - 0)^2} = \sqrt{(-6)^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}\]For \(DA\), \((-5,3)\) and \((-2,9)\):\[ DA = \sqrt{(-2 + 5)^2 + (9 - 3)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\]

Step 2: Check the Diagonals AC and BD

Now, calculate the lengths of the diagonals using the distance formula to ensure they are equal.For \(AC\), \((-2,9)\) and \((1,0)\):\[ AC = \sqrt{(1 - (-2))^2 + (0 - 9)^2} = \sqrt{3^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}\]For \(BD\), \((4,6)\) and \((-5,3)\):\[ BD = \sqrt{(-5 - 4)^2 + (3 - 6)^2} = \sqrt{(-9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}\]

Step 3: Confirm Square Properties

For the quadrilateral to be a square, all sides must be equal and both diagonals must be equal. From the previous steps, we've found:- \(AB = BC = CD = DA = 3\sqrt{5}\)- \(AC = BD = 3\sqrt{10}\)Since all sides are equal and the diagonals are equal, confirming both diagonals are longer than the sides verifies that the quadrilateral is a square.

Key Concepts

Distance Formula

The distance formula is a fundamental concept in geometry that helps us find the distance between two points on the coordinate plane. It is derived from the Pythagorean Theorem and can be applied when the coordinates of the points are known.
The distance formula is expressed as:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

To calculate the distance, identify the coordinates of the two points, subtract the x-values from each other, square the result, and do the same for the y-values. Add these squared differences together and find the square root of this sum. This formula is particularly useful in determining the lengths of sides in various shapes and for checking whether a shape is a certain type, like a square, by verifying equidistant properties.

Quadrilateral

A quadrilateral is a geometric shape with four sides and four vertices. Based on the length of its sides and internal angles, quadrilaterals can be classified as squares, rectangles, trapezoids, and more. Understanding the properties of quadrilaterals is fundamental in geometry.
In any quadrilateral:

• The sum of the internal angles is always 360 degrees.
• The shape can vary widely, from efficient parallelograms to irregular four-sided polygons.

Quadrilaterals are everywhere in geometry and are essential for studying lines, angles, and area. Knowing how to manipulate and calculate their characteristics using coordinate points is key to solving many geometric problems.

Vertices

Vertices are crucial in geometry as they represent the points where two or more line segments meet. In the context of a quadrilateral, there are four vertices, and knowing their coordinates allows for precise calculations of distance and angle measurements.
For a shape like a square, each vertex connects two sides of equal length, which can be confirmed using the distance formula. When coordinates are given, marking these points on a graph can help visualize the shape and better understand its properties.
Through vertices:

• We determine the structure and symmetry of geometric shapes.
• In quadrilaterals, they give us the basis to apply various geometric formulas.

These seemingly simple points hold the key to unlocking a detailed analysis of geometric figures.

Square

A square is a special type of quadrilateral distinguished by having four equal sides and four right angles. Verifying a square by evaluating vertices and side lengths is a practical application of the distance formula and the properties of quadrilaterals.
Key properties of a square include:

• All sides are of equal length.
• The diagonals are also equal, which further confirms its properties.
• The diagonals intersect at a right angle and are longer than the sides, offering a clear marker of its square nature.

Understanding these aspects is crucial in dissecting geometric problems that involve squares, whether checking for parallel lines, equal angles, or symmetrically opposing sides.