Problem 86 - Elementary and 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 right-angled as \((\sqrt{26})^2 + (\sqrt{26})^2 = (\sqrt{52})^2\).

Step 1: Identify the vertices

The vertices of the triangle are given as \((2, 3)\), \((-3, 4)\), and \((1, -2)\). Assign these to points \(A\), \(B\), and \(C\) respectively for convenience.

Step 2: Apply the Distance Formula

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the lengths of the sides. Calculate \(AB\), \(BC\), and \(AC\).\[ AB = \sqrt{((-3) - 2)^2 + (4 - 3)^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \]\[ BC = \sqrt{(1 - (-3))^2 + ((-2) - 4)^2} = \sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \]\[ AC = \sqrt{(1 - 2)^2 + ((-2) - 3)^2} = \sqrt{(-1)^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

Step 3: Check the Pythagorean Theorem

For the triangle to be right-angled, the squares of the lengths of two sides must add up to the square of the length of the third side. Identify the longest side, which is \(BC\) with length \(\sqrt{52}\), and check:\[ AC^2 + AB^2 = (\sqrt{26})^2 + (\sqrt{26})^2 = 26 + 26 = 52 \]Since \(52 = 52\), the Pythagorean theorem holds, confirming the triangle is right-angled.

Key Concepts

Distance Formula Application

The distance formula is a crucial tool in coordinate geometry, particularly useful when you want to find the distance between two points. Imagine you have two points on a coordinate plane with coordinates

• Point 1: ewline oindent ext (x_1, y_1)
• Point 2: ewline oindent ext (x_2, y_2)

oindent You calculate the distance between these points using the distance formula:ewline oindent \[ ext{d} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula comes from the Pythagorean theorem applied in a coordinate plane scenario. By forming a right triangle where the line connecting both points is the hypotenuse, you measure horizontal and vertical distances to form the other sides of the triangle. It computes lengths directly from point coordinates, simplifying geometry in a plane.

Understanding Right Triangles

A right triangle is a special type of triangle that always includes one angle measuring exactly 90 degrees, marked as a right angle. The sides forming the right angle are called the legs, while the side opposite the right angle is the hypotenuse, the longest side of the triangle. Right triangles are significant because they allow for the use of the Pythagorean theorem, a formula used to relate the lengths of the sides of the triangle. The theorem is stated as:\[ ext{The hypotenuse is the square root of the sum of the squares of the legs:} ewline oindent ext{c}^2 = ext{a}^2 + ext{b}^2\]This relationship is not just crucial for geometric problems but also for working out distances and angles, understanding trigonometric functions, and solving real-world applications.

Diving into Coordinate Geometry

Coordinate geometry merges algebra and geometry, allowing you to analyze geometric figures using a coordinate plane. It introduces a structured way to deal with elements like points and lines, using pairs of numerical coordinates. For example, if you have a triangle on a coordinate plane, each vertex is identified uniquely by its x and y coordinates. Coordinate geometry simplifies the task of proving properties about shapes, such as whether a triangle is right-angled. You check the measurements using coordinates and can then confirm properties using the distance formula or examining slopes. It's an efficient method for bridging algebraic techniques with geometrical visualizations. From simple line slopes to identifying intersections, coordinate geometry provides a way to descriptively tag geometric views with actual numbers, making geometry more tangible.