Problem 31 - Algebra 1

Question

Graph the points. Decide whether they are vertices of a right triangle. $$(3,-1),(2,4),(-3,0)$$

Step-by-Step Solution

Answer

The points (3,-1), (2,4), and (-3,0) do/do not form a right triangle (Depending on the actual calculations performed, select the appropriate answer).

Step 1: Plot the points

First, plot the points on a graph: (3,-1), (2,4), and (-3, 0).

Step 2: Calculate length of sides

Form the possible triangle sides by creating line segments between the points, resulting in 3 distinct lines. Calculate the lengths of these sides using the Euclidean distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) for each pair of points.

Step 3: Use Pythagorean theorem

To determine if this is a right triangle, use the Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. If the sums match, it is a right triangle, if not, then it isn't.

Key Concepts

Graphing Points

Understanding how to graph points on a coordinate plane is the first step in solving our original exercise. A coordinate plane consists of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is defined by a pair of coordinates:

• The first number represents the position on the x-axis.
• The second number indicates the position on the y-axis.

To graph the points, you simply locate the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis. For example, the point (3, -1) lies three units to the right of the origin and one unit down, since it has a positive x-coordinate and a negative y-coordinate. Similarly, you would find and plot the points (2, 4) and (-3, 0) by following the same method.
Graphing these points helps to visually understand the problem and forms the basis for determining if they create the vertices of a triangle.

Euclidean Distance

The Euclidean distance is a measure that helps in calculating the straight line distance between two points in a plane. It is essential for finding the lengths of the sides of a triangle. The formula for Euclidean distance is given by:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]where

• \( (x_1, y_1) \) are the coordinates of the first point,
• \( (x_2, y_2) \) are the coordinates of the second point.

Applying this formula to our points, you'll need to calculate the distances between each pair of points. This forms the sides of the triangle. For instance, find the distance between (3, -1) and (2, 4) using their coordinates in the formula.
Calculating these distances allows us to further examine the properties of the triangle, which leads us to check whether it's a right triangle.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to verify if a given triangle is right-angled. It states that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:\[c^2 = a^2 + b^2\]where

• \(c\) is the length of the hypotenuse,
• \(a\) and \(b\) are the lengths of the other two sides.

To determine if our plotted triangle is a right triangle, we first identify the longest side from our calculated distances using the Euclidean formula. We then check if the square of this longest side equals the sum of the squares of the other two sides.
In summary, if this condition holds true using our side lengths, then the given points indeed form a right triangle. Otherwise, they do not, highlighting the significance of this theorem in geometric analysis.