Problem 74 - Intermediate Algebra

Question

Explain how the distance formula and the Pythagorean theorem can be used to show that a triangle with vertices \((2,3),(-3,4)\) and \((1,-2)\) is a right triangle.

Step-by-Step Solution

Answer

The triangle is a right triangle with sides \( \sqrt{26} \), \( \sqrt{26} \), and \( \sqrt{52} \). The Pythagorean Theorem is satisfied.

Step 1: Understanding the Distance Formula

The distance formula is derived from the Pythagorean theorem and is used to calculate the distance between two points in a plane. If the points are \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is given by:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]We need to calculate the lengths of all three sides of the triangle using these points.

Step 2: Calculate Side Lengths Using Distance Formula

Calculate the distance between each pair of points:1. Between \((2, 3)\) and \((-3, 4)\): \[ AB = \sqrt{(-3 - 2)^2 + (4 - 3)^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \]2. Between \((-3, 4)\) and \((1, -2)\): \[ BC = \sqrt{(1 - (-3))^2 + (-2 - 4)^2} = \sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \]3. Between \((2, 3)\) and \((1, -2)\): \[ CA = \sqrt{(1 - 2)^2 + (-2 - 3)^2} = \sqrt{(-1)^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

Step 3: Use the Pythagorean Theorem

The Pythagorean 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. We need to check whether \(AB^2 + CA^2 = BC^2\).\[ AB^2 = (\sqrt{26})^2 = 26 \]\[ CA^2 = (\sqrt{26})^2 = 26 \]\[ BC^2 = (\sqrt{52})^2 = 52 \]Check:\[ AB^2 + CA^2 = 26 + 26 = 52 = BC^2 \]

Step 4: Conclusion

Since \(AB^2 + CA^2 = BC^2\), it satisfies the condition of the Pythagorean theorem confirming that the triangle is indeed a right triangle with the right angle at vertex \((2, 3)\),\((-3,4)\), and \((1,-2)\).

Key Concepts

The Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates to right triangles. It 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. This can be mathematically expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

This theorem is immensely useful for verifying if a triangle is right-angled. By calculating the side lengths and checking if the formula holds, you can confirm the presence of a right angle. This principle was pivotal in solving the exercise by showing that the sum of the squares of two sides equaled the square of the third side, confirming a right triangle.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry using a coordinate plane. In this method, points on the plane are defined by the coordinates \((x, y)\). The distance formula, derived using the Pythagorean theorem, plays a crucial role here. It calculates the straight-line distance between any two points in the plane, which was essential in this exercise.

To determine if the triangle is right-angled, the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) was used to find the lengths of each side of the triangle by substituting the respective coordinates. By linking geometric figures with algebraic equations, coordinate geometry facilitates the solution of complex problems involving shapes and distances, as it did here.

Right Triangle

Right triangles are a special classification of triangles that include one angle exactly equal to 90 degrees. This special angle significantly influences the properties and calculations involving the triangle. One of their most recognizable properties involves the Pythagorean theorem. In our exercise, proving the triangle was right-angled relied on leveraging the characteristics of right triangles and applying them to the side lengths calculated.

One identifying feature of right triangles, relevant to the exercise, is that they can be easily analyzed using the distance formula. If two squares of the side lengths sum up to the square of the third side, as shown, the triangle is certainly a right triangle. Recognizing its right-angle status allows for straightforward problem-solving and application of mathematical principles.