Problem 36 - Algebra 2

Question

OPEN ENDED Find two points that are \(\sqrt{29}\) units apart.

Step-by-Step Solution

Answer

Points (0,0) and (5,2) or (0,0) and (-5,-2) or (0,0) and (2,5) are \\(\sqrt{29}\\) units apart.

Step 1: Understand the distance formula

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) on a coordinate plane is calculated using the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \(d\) is the distance between the points.

Step 2: Set up the equation

We are given that the distance \(d = \sqrt{29}\). We need to find two points such that when substituted in the distance formula, the distance equals \(\sqrt{29}\). So, the equation becomes: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{29} \).

Step 3: Simplify the equation

Square both sides to eliminate the square root, resulting in: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 = 29 \). Now, we need to find integer combinations for \(x_1, y_1, x_2,\) and \(y_2\) that satisfy this equation.

Step 4: Try possible integer points

To simplify our task, let's choose a point, say \(x_1 = 0, y_1 = 0\) (origin) for simplicity, and find \(x_2, y_2\) such that \( x_2^2 + y_2^2 = 29 \). Look for integer solutions to this equation.

Step 5: Check potential pairs

Try integer values systematically. After checking, one possible solution is \(x_2 = 5, y_2 = 2\) which gives: \( 5^2 + 2^2 = 25 + 4 = 29 \). Another option by symmetry is \(x_2 = -5, y_2 = -2\) or \(x_2 = 2, y_2 = 5\) and their negatives.

Step 6: Conclude valid points

The pairs of points that are \(\sqrt{29}\) units apart are (0, 0) and (5, 2). Alternatively, you can use (0, 0) and (-5, -2) or (0, 0) and (2, 5) or the negatives (0, 0) and (-2, -5) are also solutions.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to represent mathematical concepts. It consists of two perpendicular axes: the horizontal axis, known as the x-axis, and the vertical axis, called the y-axis. These axes intersect at the origin, denoted as the point (0, 0).

Each point on this plane is defined by a pair of coordinates, \(x, y\), which show its position relative to the origin. The coordinate plane allows us to visualize and solve problems such as locating points, finding distances between them, and analyzing geometric shapes. This visual representation makes understanding and working with algebraic expressions much simpler and concrete.

Using the coordinate plane is essential for applying formulas like the distance formula, as it assigns numerical values to the geometric positions of the points we are examining.

Integer Solutions

Integer solutions refer to solutions of equations where the values are whole numbers. When working with coordinates on a plane and tasks like finding two points a specific distance apart, integer solutions provide clear, precise answers without involving decimals or fractions.

Finding integer solutions requires evaluating various combinations of whole numbers for the coordinates. This involves substituting potential coordinates into the given equation to determine if they satisfy the conditions of the problem. In the context of the exercise we discussed, the task was to find integers \(x_2, y_2\) such that \(x_2^2 + y_2^2 = 29\).

Trying different integers methodically and using symmetry can simplify the process to find correct solutions. For example, testing \(x = 5, y = 2\) gives a solution since \(5^2 + 2^2 = 25 + 4 = 29\). Integer solutions are significant in ensuring that the solutions are both simple and reliable, providing clear answers for mathematical problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem is often represented by the equation \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the triangle's legs, and \(c\) is the length of the hypotenuse. This principle is not only pivotal in solving geometrical problems but also serves as the foundation for the distance formula used in coordinate geometry.

In the context of our exercise, the Pythagorean Theorem helps establish the connection between algebraic equations and geometric concepts. By setting \( d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2\), we use the theorem to find distances on a coordinate plane. The accuracy and reliability of the Pythagorean Theorem make it a powerful tool for both theoretical and applied mathematics, including finding points at a specified distance apart.