Problem 32 - Algebra 1

Question

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

Step-by-Step Solution

Answer

After calculating the lengths of the sides, apply the Pythagorean theorem to determine if it’s a right triangle. The answer depends on whether or not the theorem holds true.

Step 1 Plot the Points on the Graph

Plot the given points \(-2,2\), \(3,4\), and \(4,2\) on the graph. Join these points to form a triangle.

Step 2 Calculate the Lengths of the Sides of the Triangle

Calculate the lengths of the sides of the triangle using the formula for the distance between two points, which is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3 Apply the Pythagorean theorem

Once the lengths are calculated, identify the longest side. Then apply the Pythagorean theorem to see if the square of the longest side is equal to the sum of the squares of the other two sides. If it is, then the triangle is a right triangle.

Key Concepts

Distance Formula

The Distance Formula is a vital tool in coordinate geometry, used to calculate the distance between two points on a plane. This formula arises from the Pythagorean Theorem and is given by: \[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]It helps connect algebra and geometry by offering a way to find lengths between points based on their coordinates. To use this formula, simply follow these steps:

• Start by identifying the coordinates of the two points you're interested in.
• Subtract the x-coordinate of the first point from the x-coordinate of the second point.
• Subtract the y-coordinate of the first point from the y-coordinate of the second point.
• Square both of these results, then sum them together.
• Finally, take the square root of the sum to find the distance.

In the given exercise, to find the lengths of the sides of the triangle formed by the points \((-2,2)\), \((3,4)\), and \((4,2)\), apply the Distance Formula to each pair of points. Once you have all three sides, you can move to the next step of verifying whether this triangle is a right triangle.

Pythagorean Theorem

The Pythagorean Theorem is a powerful mathematical principle applied in right triangles. The theorem states that for any right triangle, the square of the length of the hypotenuse (the triangle's longest side) equals the sum of the squares of the lengths of the other two sides. Formally, this is expressed by the equation:\[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. This theorem is particularly useful in coordinate geometry for proving whether a given triangle is a right triangle.In the context of the exercise's problem, once you calculate the lengths of all sides using the Distance Formula, identify the longest side. Check if the sum of the squares of the two shorter sides equals the square of the longest side. If they do, the triangle is indeed a right triangle. Thus, the Pythagorean Theorem provides a simple verification method after computing the side lengths.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, allows us to discuss or prove geometric principles using the coordinate plane and algebra. It combines the fields of geometry and algebra. This branch of geometry provides a concrete method of finding distances between points, midpoints, and the slopes of lines, enhancing our geometric understanding with algebraic techniques.In the given exercise, coordinate geometry enables us to graphically represent the points \((-2,2)\), \((3,4)\), and \((4,2)\). After plotting the points on a coordinate plane, we create a visual representation that helps us understand the geometric relationships between these points.Key benefits of using coordinate geometry include:

• Visualizing complex geometric situations effortlessly.
• Applying algebraic methods to solve geometric problems.
• Determining properties like parallelism, perpendicularity, and collinearity of lines.

By setting the triangle within the coordinate plane, students can better appreciate how these mathematical areas interact, witnessing firsthand how plotting points can lead to deeper understanding of geometric principles. This exercise becomes a pivotal example of finding vertices of a right triangle through a blend of geometric intuition and algebraic computation.