Equation of the Sides of an Equilateral Triangle with a Given Vertex

In this article, we will explore how to find the equations of the sides of an equilateral triangle when given one of its vertices and the base lying on the x-axis. We'll break down the problem into manageable steps and provide the necessary mathematical derivations.

1. Understanding the Problem

Given a vertex at point (-12, 2) and a base along the line y 0, we need to find the equations of the sides of the equilateral triangle.

2. Step-by-Step Solution

Step 1: Determine the Length of the Sides

In an equilateral triangle, all sides are of equal length. Let's denote the side length as s. The height h from the vertex (-12, 2) to the base y 0 is 2. The relationship between the height h and the side length s in an equilateral triangle is given by:

h (sqrt{3}/2) * s

Step 2: Solve for the Side Length s

From the height equation:

2 (sqrt{3}/2) * s

Solving for s:

s (2 * 2) / sqrt{3}

Approximately:

s ≈ 2.31

Step 3: Find the Coordinates of the Other Two Vertices

The base of the triangle is along the line y 0. Since the triangle is equilateral, the two base vertices will be symmetrically positioned around the vertical line through the vertex (-12, 2). Let’s denote the base vertices as A and B. The midpoint of the base is directly below (-12, 2) at (-12, 0). Thus, the distance from this midpoint to each base vertex is:

Shift s / 2 (4 / sqrt{3}) / 2 2 / sqrt{3}

Approximately:

Shift ≈ 1.15

The coordinates of the base vertices A and B will be:

A ≈ (-12 - 1.15, 0) ≈ (-13.15, 0) B ≈ (-12 1.15, 0) ≈ (-10.85, 0)

Step 4: Equations of the Sides

Now, we can find the equations of the lines forming the sides of the triangle. The three sides are:

Side from vertex (-12, 2) to vertex A (~-13.15, 0)

- Slope m_1 (0 - 2) / (-13.15 - (-12)) -2 / -1.15 ≈ 1.74

- Equation: y - 2 1.74(x 12)

Approximately:

y 1.74x 3.74

Side from vertex (-12, 2) to vertex B (~-10.85, 0)

- Slope m_2 (0 - 2) / (-10.85 - (-12)) -2 / 1.15 ≈ -1.74

- Equation: y - 2 -1.74(x 12)

Approximately:

y -1.74x - 0.26

Base from A to B

Since both points are on the line y 0, the equation is simply:

y 0

5. Drawing the Figure

To draw the figure of the equilateral triangle:

Plot the vertex at (-12, 2) Plot the base vertices A (~-13.15, 0) and B (~-10.85, 0) Draw the line segments connecting (-12, 2) to A and B, and the line segment connecting A and B along y 0

Summary of the Equations

Side from (-12, 2) to A (~-13.15, 0):

y 1.74x 3.74

Side from (-12, 2) to B (~-10.85, 0):

y -1.74x - 0.26

Base: y 0

This gives you a complete representation of the equilateral triangle with the specified vertex and base.