Problem 26 - Algebra 1: Concepts and Skills

Question

Graph the points. Determine whether they are vertices of a right triangle. $$ (5,4),(2,1),(-3,2) $$

Step-by-Step Solution

Answer

By plotting the points on graph paper, calculating the distances between each pair, and checking if those distances satisfy the Pythagorean theorem, we can determine if the given points are vertices of a right triangle.

Step 1: Plotting the Points

Plot the given points (5,4),(2,1),(-3,2) on the Cartesian plane.

Step 2: Calculating Euclidean Distances

Calculate the Euclidean distance between each pair of points to determine the lengths of the triangle's sides. The formula for the distance between two points \((x1, y1)\) and \((x2, y2)\) is \( \sqrt{(x2-x1)^2 + (y2-y1)^2} \). So, calculate the distance between points (5,4) and (2,1), points (5,4) and (-3,2), and points (2,1) and (-3,2).

Step 3: Applying Pythagorean Theorem

Use the Pythagorean theorem, \(a^2 + b^2 = c^2\), where c is the length of the hypotenuse and a and b are the lengths of the other two sides, to check if the points form a right triangle. If they satisfy the equation, then they indeed form a right triangle.

Key Concepts

Cartesian Plane

The Cartesian Plane, or coordinate plane, is a two-dimensional surface where each point is identified by a pair of numerical coordinates. These coordinates determine the position of the point on the plane relative to the horizontal and vertical axes.

• The horizontal axis is known as the x-axis.
• The vertical axis is called the y-axis.

Each point on this plane is represented by an ordered pair \(x, y\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
To plot a point like (5,4) on the Cartesian Plane:

• Start at the origin \( (0,0) \).
• Move 5 units in the positive direction along the x-axis.
• Then move 4 units up in the positive direction along the y-axis.

Plotting points allows us to visualize geometric figures and analyze their properties, such as whether they form a specific type of triangle.

Euclidean Distance

Euclidean Distance is a measure of the "straight line" distance between two points in the Cartesian Plane. It is derived from the Pythagorean Theorem and is the most commonly used distance metric in geometry.
The formula for finding the distance \( d \) between two points \( (x1, y1) \) and \( (x2, y2) \) is: \[ d = \sqrt{(x2-x1)^2 + (y2-y1)^2} \]

• This formula calculates the hypotenuse of a right triangle formed by the horizontal and vertical differences between two points.
• The differences \( (x2-x1) \) and \( (y2-y1) \) are the horizontal and vertical legs of the right triangle, respectively.

By applying this formula to given points, we calculate the length of the sides of the triangle determined by these points. These distances are crucial in verifying the type of triangle formed by the points.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that defines the relationship between the sides of a right triangle. According to this theorem, in a right triangle with sides \(a\), \(b\), and the hypotenuse \(c\), we have:\[ a^2 + b^2 = c^2 \]The hypotenuse is the longest side of the right triangle, opposite the right angle.
This theorem helps verify if a set of three points forms a right triangle:

• Calculate the distances between each pair of points to get the lengths of the sides.
• Identify the longest side; this should be \(c\).
• Check if the sum of the squares of the other two sides equals the square of \(c\).

If this equation holds, then the triangle is a right triangle. This logical approach allows us to determine right triangles just from coordinate points.