Question

Algebra \(\triangle D E F\) is an equilateral triangle in which \(E D=x+5\), \(D F=3 x-3\), and \(E F=2 x+1\). a. Draw and label \(\triangle D E F\). b. Find the measure of each side.

Short answer

Each side of triangle \(\triangle DEF\) measures 9 units.

Step-by-step solution

Understanding the problem

We are given an equilateral triangle \(\triangle DEF\) with sides \(ED = x + 5\), \(DF = 3x - 3\), and \(EF = 2x + 1\). An equilateral triangle means all sides are equal in length.

Set up equations for equal sides

Since \(\triangle DEF\) is equilateral, we equate the expressions for the sides:1. \(x + 5 = 3x - 3\)2. \(x + 5 = 2x + 1\)3. \(3x - 3 = 2x + 1\) We need to solve any of these pairs to find the value of \(x\).

Solve Equations

Let's solve the equation \(x + 5 = 3x - 3\). Subtract \(x\) from both sides to get: \(5 = 2x - 3\).Add 3 to both sides: \(8 = 2x\).Divide by 2: \(x = 4\).

Verify other equations

Substituting \(x = 4\) into the second equation \(x + 5 = 2x + 1\) gives \(4 + 5 = 2(4) + 1\), which simplifies to \(9 = 9\), confirming it holds true. Testing the third equation \(3x - 3 = 2x + 1\), gives \(3(4) - 3 = 2(4) + 1\), simplifying to \(9 = 9\), which also holds true.

Calculate the sides

Substitute \(x = 4\) back into the expressions for each side:- \(ED = x + 5 = 9\)- \(DF = 3x - 3 = 9\)- \(EF = 2x + 1 = 9\)Thus, each side of the triangle \(\triangle DEF\) measures 9 units.

Key concepts

Algebraic Equations

Algebraic equations are fundamental in solving problems involving unknown variables. They are mathematical statements where two expressions are set equal to each other. In the case of an equilateral triangle, like \( \triangle DEF \), algebraic equations help us express the equality of the sides using expressions provided.
Understanding how to set up these equations relies on the properties of the shape involved. We know \( \triangle DEF \) is equilateral, meaning all its sides are equal. Therefore, if the expressions for the sides are given as \( E D = x + 5 \), \( D F = 3x - 3 \), and \( E F = 2x + 1 \), setting these equal to each other forms the basis of our algebraic equations. Here, we create three primary equations:

• \( x + 5 = 3x - 3 \)
• \( x + 5 = 2x + 1 \)
• \( 3x - 3 = 2x + 1 \)

These equations represent the balance needed to maintain the equilateral property and will help us solve for the unknown variable \( x \).

Solving for Variables

Solving for variables is a critical skill in algebra which involves finding the value of unknown quantities represented by variables. In our exercise with \( \triangle DEF \), we solve the equation \( x + 5 = 3x - 3 \) to find \( x \).
Here's a simple breakdown:

• Subtract \( x \) from both sides to get \( 5 = 2x - 3 \).
• Add 3 to both sides resulting in \( 8 = 2x \).
• Finally, divide by 2 to solve for \( x \), yielding \( x = 4 \).

Plugging back \( x = 4 \) into the other equations serves as a verification step to ensure consistency. Through substitution, it's confirmed that each side evaluates correctly, hence proving \( x = 4 \) is indeed the right solution.

Equilateral Triangle Properties

Equilateral triangles are unique and notable for their symmetry, which simplifies many geometric calculations. A crucial property of an equilateral triangle is that all its sides are of equal length. Also, all internal angles are equal, each measuring 60 degrees.
The problem with \( \triangle DEF \) illustrates these properties effectively. Since it's equilateral, once we determine one side, we automatically know the lengths of all three sides. After solving the algebraic equations, we found each side equaled 9 units, perfectly aligning with equilateral characteristics.
This exercise not only reinforces the property of side equality but also shows how knowing one characteristic of an equilateral triangle simplifies finding other measures. This symmetry and equality make equilateral triangles a favorite subject in geometry to explore various mathematical concepts.