Problem 27 - Algebra 1

Question

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

Step-by-Step Solution

Answer

By calculating the distances between the points, applying these distances to the Pythagorean theorem and checking for compliance, we can determine whether the given points form a right triangle or not. Whether they do indeed form such a triangle will be revealed by the calculations and cannot be stated without.

Step 1: Plot the points

Plot the points \( (5,4), (2,1), \) and \( (-3,2) \) on the Cartesian coordinate system to represent the potential vertices of the triangle.

Step 2: Calculate distances

Use the distance formula, \(d=\sqrt{{(x_2-x_1)^2+(y_2-y_1)^2}}\), to calculate the distances between all three pairs of points. That is, the distances between the points (5,4) & (2,1), (2,1) & (-3,2), and (-3,2) & (5,4). These distances represent the potential sides of the triangle.

Step 3: Apply the Pythagorean theorem

With the three distances calculated, label the longest distance as 'c'. The other two shorter distances will be labelled as 'a' and 'b'. Apply the Pythagorean Theorem formula \(a^2 + b^2 = c^2\) to check if these distances satisfy the theorem. The triangle will be a right triangle if the formula is satisfied, i.e., if \(a^2 + b^2\) equals to \(c^2\) or ca be approximated to it.

Key Concepts

Distance Formula

The distance formula is a crucial tool in geometry that helps us determine the length between two points in the Cartesian coordinate system. Imagine two points, \((x_1, y_1)\) and \((x_2, y_2)\), on a 2D plane. The formula we use to find the distance \(d\) between these two points is:
\[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \]
This formula is essentially a restatement of the Pythagorean theorem for a coordinate grid. It calculates the hypotenuse of a right triangle formed by the difference in the x-coordinates and the y-coordinates of these two points.
Here's how it works:

• Calculate the difference between the x-coordinates \((x_2-x_1)\) and the y-coordinates \((y_2-y_1)\).
• Square both differences to ensure we're working with positive values.
• Add these squares together and then take the square root of the total.

In the context of plotting vertices to form a triangle, using the distance formula helps identify the length of each side, which is vital for further calculations, like applying the Pythagorean theorem to verify the type of triangle the points form.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially when working with right triangles. It states that in a right 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 other two sides. Mathematically, it's expressed as:
\[ a^2 + b^2 = c^2 \]
where \(c\) represents the hypotenuse and \(a\) and \(b\) are the triangle's other two sides.
When you have a set of points in a plane and need to determine if they form a right triangle, you can:

• Calculate the distances between each pair of points using the distance formula.
• Identify the longest distance as the hypotenuse \(c\) and the other two as \(a\) and \(b\).
• Check if \(a^2 + b^2 = c^2\). If it holds true, the points form a right triangle.

This theorem is incredibly useful in solving many real-world problems, including navigation, construction, and physics.

Cartesian Coordinate System

The Cartesian coordinate system is a standardized system for defining a point's location in a plane using a pair of numerical coordinates. It forms a grid defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis.
Each point on this grid is described by an ordered pair \((x, y)\), where:

• The "x" value defines the horizontal distance from the origin (0,0).
• The "y" value defines the vertical distance from the origin.

This system allows for easy plotting and visualization of geometric shapes like triangles, squares, and circles. Understanding how to navigate and utilize the Cartesian coordinate system is crucial in various fields such as physics, engineering, and computer graphics.
In problems like identifying the type of triangle formed by specific points, the Cartesian plane helps us visually inspect plots, measure distances, and apply the distance formula accurately. For the provided example, plotting the given points on this grid helps ascertain the nature of the triangle they form by simplifying the determination of distances and angles.