Problem 70 - Elementary and Intermediate Algebra

Question

Use the distance formula to show that a triangle with vertices \((-2,4),(2,8),\) and \((6,4)\) is isosceles.

Step-by-Step Solution

Answer

The triangle with the given vertices is isosceles because it has two equal sides of length \(4\sqrt{2}\).

Step 1: Identify the Distance Formula

The distance formula is used to calculate the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane. It is given by the formula: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Step 2: Calculate Distance Between First Two Vertices

Use the distance formula to calculate the distance between points \((-2, 4)\) and \(2, 8)\).\[D = \sqrt{(2 - (-2))^2 + (8 - 4)^2} = \sqrt{(2 + 2)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]So, the distance between the first two vertices is \4\sqrt{2}\.

Step 3: Calculate Distance Between Second and Third Vertices

Use the distance formula to calculate the distance between points \(2, 8)\) and \(6, 4)\).\[D = \sqrt{(6 - 2)^2 + (4 - 8)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]The distance between the second and third vertices is \4\sqrt{2}\.

Step 4: Calculate Distance Between First and Third Vertices

Use the distance formula to calculate the distance between points \((-2, 4)\) and \(6, 4)\).\[D = \sqrt{(6 - (-2))^2 + (4 - 4)^2} = \sqrt{(6 + 2)^2 + 0^2} = \sqrt{8^2} = 8\]The distance between the first and third vertices is \8\.

Step 5: Assess Triangle for Isosceles Property

A triangle is isosceles if at least two of its sides are of equal length. From our calculations, the distances are:- Between the first and second vertices: \4\sqrt{2}\.- Between the second and third vertices: \4\sqrt{2}\.- Between the first and third vertices: \8\.Since the distances between the first and second vertices and the second and third vertices are equal, the triangle with vertices \((-2,4)\), \(2,8)\), and \(6,4)\) is isosceles.

Key Concepts

Isosceles Triangle

An isosceles triangle is a special type of triangle with two sides of equal length. These equal sides are called the 'legs,' while the third side is known as the 'base.' One of the defining properties of an isosceles triangle is that it also has two angles of equal measure opposite the equal sides. These angles are called the 'base angles.'

• Equal side lengths
• Two angles of equal measure
• Symmetrical structure

In the exercise given, the calculated distances between the triangle's vertices confirm that two sides are equal, proving it's isosceles. Recognizing these properties not only helps in geometry but also serves well in coordinate geometry where we use calculations to prove them.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using a coordinate plane. This topic is all about using mathematical equations and formulae to determine distances, angles, midpoints, and other properties of shapes formed by points defined by their coordinates.

• Point representation using coordinates
• Utilization of the distance formula for length calculations
• Analysis of geometric figures using calculations

In our triangle exercise, applying coordinate geometry enables us to ascertain properties like side lengths by calculating the distances between points. By verifying these calculations, students can clearly see how the position of points in the coordinate system translates into specific geometric properties.

Triangle Properties

Triangles, fundamental shapes in geometry, have unique properties based on the lengths of their sides and magnitude of their angles.

• Scalene: All sides are of different lengths.
• Isosceles: Two sides are equal.
• Equilateral: All sides are equal and all angles measure 60 degrees.
• Sum of interior angles equals 180 degrees.

Each type exhibits specific properties which can be determined through calculations and logical reasoning. In the case of the isosceles triangle with given vertices, the assignments of these properties stem from the equivalent distances calculated using the distance formula, reinforcing the foundational geometric concepts.