Problem 30 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

By plotting the points, drawing the triangle, calculating the lengths of each side, and applying the Pythagorean theorem, it can be determined that the points (3,-1), (2,4), and (-3,0) are indeed vertices of a right-angled triangle.

Step 1: Plot the points on the graph

Begin by plotting the given points on a graph. The points are (3,-1), (2,4), and (-3,0). Each point corresponds to an (x,y) value on the graph.

Step 2: Identify the triangle

Once the points are plotted, a triangle should be identifiable. The triangle's vertices are the provided points.

Step 3: Measure distances between vertices

To determine if the triangle is a right triangle, we measure the distances from each point to the other two points using the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) for each pair of points.

Step 4: Apply the Pythagorean theorem

Now, check if the square of the longest side equals the sum of the squares of the other two sides. If they are equal, then by the Pythagorean theorem, it is a right-angled triangle.

Key Concepts

Graphing Points

Graphing points is a fundamental step in analyzing any geometric problem. When given coordinates like

• (3, -1)
• (2, 4)
• (-3, 0)

you plot them on a Cartesian plane, often labeled with an x-axis (horizontal) and a y-axis (vertical). Each coordinate pairs an x-value with a y-value, representing a unique position on the graph.
To correctly plot these points:

• Locate the x-coordinate on the horizontal axis.
• Then, find the corresponding y-coordinate on the vertical axis.
• Finally, mark the point where these two coordinates intersect.

This step transforms abstract numbers into a visual representation, making it easier to see the geometric relationships between points.
In this exercise, plotting (3,-1), (2,4), and (-3,0) will form the vertices of a potential triangle. This visualization helps in verifying if the given points form any special triangle type, such as a right triangle.

Distance Formula

The distance formula helps find the length between two points in the coordinate plane. It's derived from the Pythagorean theorem and is written as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula calculates the straight-line distance between any two points

• (x₁, y₁)
• (x₂, y₂)

on the plane.
To apply the distance formula, subtract the x-values (\(x_2 - x_1\)) and the y-values (\(y_2 - y_1\)) of your points. Square the results, add them together, and finally take the square root.
For example, to find the distance between points (3, -1) and (2, 4), substitute these into the formula:\[ d = \sqrt{(2 - 3)^2 + (4 - (-1))^2} = \sqrt{1 + 25} = \sqrt{26} \]Repeat this process for the other pairs to find the lengths of all sides. Understanding these lengths is crucial as it will determine whether these points form a right triangle when checked with the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is essential in determining whether a triangle is a right triangle. It states that for a right triangle:\[ c^2 = a^2 + b^2 \]Here, "c" represents the hypotenuse (the longest side), while "a" and "b" are the other two shorter sides.
After using the distance formula to find each side of your triangle, identify the longest side.
Check if the square of this longest side equals the sum of the squares of the other two sides.

• If they match, the points form a right triangle.
• If not, the triangle is not a right triangle.

This simple yet powerful theorem allows us to confirm right-angled triangles reliably, making it one of the cornerstone principles in geometry.