Problem 43 - Precalculus with Limits

Question

In Exercises 43-46, show that the points form the vertices of the indicated polygon. Right triangle: \( (4, 0) \), \( (2, 1) \), \( (-1, -5) \)

Step-by-Step Solution

Answer

If the calculated distances satisfy the Pythagorean theorem, the points (4,0), (2,1), and (-1,-5) form a right triangle.

Step 1: Calculation of the Lengths of Sides

First, calculate the distances between each pair of points to find out the length of each side of the triangle. Use the Euclidean distance formula which is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). For example, to calculate the distance between (4,0) and (2,1) apply the formula and find the length \(d_{12}\). Do this for all three pairs of points to get \(d_{12}\), \(d_{13}\), \(d_{23}\).

Step 2: Application of Pythagorean theorem

Sort the calculated distances in increasing order. Apply the Pythagorean theorem on these sorted distances. If the equation holds, the given points form a right triangle. If not, they do not form a right triangle.

Key Concepts

Euclidean Distance Formula

The Euclidean Distance Formula is a technique used to calculate the distance between two points in a coordinate plane. It's a key tool in geometry and helps determine the length between points on a graph. To find the distance between two points \(x_1, y_1\) and \(x_2, y_2\), you use the formula:

• \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

This formula stems from the Pythagorean Theorem and effectively measures the diagonal of a right triangle formed by the grid lines between the points. Let's take some example points: (4,0) and (2,1).
Plug them into the formula as follows:

• \( \sqrt{(2-4)^2 + (1-0)^2} \)
• \( \sqrt{(-2)^2 + 1^2} \)
• \( \sqrt{4 + 1} = \sqrt{5} \)

So, the distance is \( \sqrt{5} \).

Use this formula for each pair of points to find the lengths of all sides.

Vertices

Vertices are the corner points of geometric shapes, like polygons. In a triangle, the vertices are where the sides meet. For example, in our exercise, the points (4, 0), (2, 1), and (-1, -5) are the vertices of the polygon we're examining.

• Each pair of vertices forms a side of the polygon.
• Understanding where vertices are located helps in determining the shape's properties.

To determine the nature of the polygon, calculate the distance between these vertices using the Euclidean Distance Formula. This helps confirm the type of triangle or polygon, such as determining whether it’s a right triangle.

Pythagorean Theorem

The Pythagorean Theorem is vital in determining if a triangle with calculated sides is a right triangle. The theorem 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 lengths of the other two sides.
• Mathematically, it is expressed as \(a^2 + b^2 = c^2\).

Once you have calculated the lengths of the triangle's sides using the Euclidean Distance Formula, apply the theorem.
First, identify the longest side, which could be the hypotenuse (\(c\)). Next, verify if the sum of the squares of the other two sides (\(a\) and \(b\)) equals the square of this long side.
If the equation holds, the triangle is a right triangle.

Polygon

A polygon is a two-dimensional shape with straight sides. In math problems, identifying and analyzing polygons often begin with understanding the vertices and sides.

• Triangles, quadrilaterals, and pentagons are common polygons.
• A right triangle is a special type of polygon with one right angle.

For the triangle given in the exercise, verify its type by checking its vertices and sides.
The set of points forms a triangle – a polygon with three sides. By using distance calculations and the Pythagorean theorem, determine if it’s a right triangle. Knowing the properties of the triangle helps classify the shape and solve further geometry problems.