Problem 44 - Precalculus with Limits

Question

In Exercises 43-46, show that the points form the vertices of the indicated polygon. Right triangle: \( (-1, 3) \), \( (3, 5) \), \( (5, 1) \)

Step-by-Step Solution

Answer

The given points form a right triangle if and only if the distances between the points satisfy the Pythagorean theorem. By calculating the distances and applying the Pythagorean theorem, it can be determined whether or not the given points form a right triangle.

Step 1: Calculating Distances between the Points

First, calculate the distances between each pair of points. This is done by using the distance formula between two points: \(d\) = \(\sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\), where (x_1, y_1) and (x_2, y_2) are two points. For \( (-1, 3) \) and \( (3, 5) \), the distance \( d_1 \) can be found as \(\sqrt{{(3 + 1)}^2 + {(5 - 3)}^2}\), for \( (3, 5) \) and \( (5, 1) \), the distance \( d_2 \) can be found as \(\sqrt{{(5 - 3)}^2 + {(1 - 5)}^2}\), and for \( (5, 1) \) and \( (-1, 3) \), the distance \( d_3 \) can be found as \(\sqrt{{(-1 - 5)}^2 + {(3 - 1)}^2}\).

Step 2: Applying the Pythagorean Theorem

Then, apply the Pythagorean theorem to these distances. An important point to remember is to use the longer distance (the hypotenuse) in the right hand side of the equation. Enumerate the distances in decreasing order and label them \( a \geq b \geq c \). If it can be shown that \( a^2 = b^2 + c^2 \), then we can confirm that the points form a right triangle.

Step 3: Verifying the Equation

Verify the result obtained from step 2. If the equation is established, then the points form a right triangle. If the equation is not established, the points do not form a right triangle.

Key Concepts

Distance Formula

Understanding the distance formula is key to solving many problems in coordinate geometry, especially when determining the lengths of sides in polygons like triangles. The distance formula \( d = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2} \) is derived from the Pythagorean theorem and calculates the straight-line distance between two points in a coordinate plane.

When applied to find the distances between the vertices of a potential right triangle, the distance formula helps pinpoint the exact lengths of the sides. To ensure clarity, let's break down an example:

• Consider the points \( (-1, 3) \) and \( (3, 5) \) on the coordinate plane. To find the distance \( d_1 \) between these points, we plug the coordinates into the formula, resulting in \( d_1 = \sqrt{{(3 + 1)}^2 + {(5 - 3)}^2} \).
• Repeating this process for the other two pairs of points, we establish the lengths of all sides of the triangle.

Using the formula accurately is crucial, as any error in calculation can lead to incorrect conclusions about the nature of the polygon in question.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, especially within the realm of geometry. It states that in a right triangle, the square of the length of the hypotenuse \( (c) \) is equal to the sum of the squares of the lengths of the other two sides \( (a) \) and \( (b) \): \( a^2 + b^2 = c^2 \).

This theorem provides a simple yet powerful test for determining whether a given set of points forms a right triangle when their coordinates are known. Once you have computed the lengths of all three sides using the distance formula:

• Arrange them in descending order, labeling them as \( a \), \( b \), and \( c \) with \( c \) being the longest side (potentially the hypotenuse).
• Apply the Pythagorean theorem to see if the equation \( a^2 = b^2 + c^2 \) holds true.

If it does, the triangle is right-angled at the vertex opposite side \( c \). It's crucial to apply the theorem correctly to avoid misjudging the type of triangle formed by the points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a system of coordinates on a plane. It unifies algebraic and geometric principles, allowing for the precise calculation of distances, angles, and other geometric properties using algebraic formulas.

In the context of identifying a right triangle, coordinate geometry involves:

• Plotting the given vertices on the coordinate plane.
• Using the distance formula to find the lengths of the sides of the triangle.
• Applying the Pythagorean theorem to determine if these lengths make up a right triangle.

Exercise Improvement Advice

To enhance the understanding of the exercise, it is helpful to visualize the triangle by sketching it on graph paper or using graphing software. Label the sides in relation to the coordinates provided, and annotate with the calculation steps. This visualization aids in comprehending why the distance formula and Pythagorean theorem are applied in a certain way and reinforces the interconnectedness of the coordinate system and geometric concepts.