91Ó°ÊÓ

Understanding how to classify triangles is an essential part of mastering geometry. Triangle classification is based on both the sides and the angles of the triangle. There are three main types of triangles based on their sides:

• **Equilateral Triangle**: All sides are equal in length, and all angles are equal, typically measuring 60 degrees each.
• **Isosceles Triangle**: Two sides are equal in length, which also means two of its angles are equal.
• **Scalene Triangle**: All sides are different in length, and consequently, all angles are also different.

The exercise requires us to classify the triangle formed by the cities of Charlotte, Raleigh, and Winston-Salem using the distances between them. By determining the lengths of these sides (70 miles, 92 miles, and 130 miles), we observe that no two sides are the same. As no side length matches any other, this triangle is a scalene triangle.

Additionally, triangles can also be classified by their angles:

• **Acute Triangle**: All angles are less than 90 degrees.
• **Right Triangle**: One angle is exactly 90 degrees.
• **Obtuse Triangle**: One angle is greater than 90 degrees.

In this case, we would calculate the angles using the side lengths, but given no side equals the sum of the squares of the other two, it’s not a right triangle.

Calculating the distance in a geometric problem like this often involves setting up relationships and solving for unknowns. Let's look at how we calculate distances here based on the problem statement. We initially introduce variables for the distances:

• \( d_1 \): The distance from Charlotte to Winston-Salem
• \( d_2 \): The distance from Winston-Salem to Raleigh
• \( d_3 \): The distance from Raleigh to Charlotte

Based on the problem's description, we define the relationships as:

• \( d_1 = d_2 - 22 \) which means "the distance from Charlotte to Winston-Salem is 22 miles less than the distance from Raleigh to Winston-Salem."
• \( d_3 = d_1 + 60 \) which implies that "the distance from Charlotte to Raleigh is 60 miles greater than the distance from Winston-Salem to Charlotte."

By using the known total distance (292 miles), which means the sum of all three distances equals this amount, we set up the equation \( d_1 + d_2 + d_3 = 292 \). Substituting the expressions for \( d_1 \) and \( d_3 \) allows us to find each specific distance.

Equation solving is crucial in unraveling the distances in this triangle problem. After substituting expressions from the relationships described, the equation
\[(d_2 - 22) + d_2 + (d_2 + 38) = 292\]
simplifies into the linear equation
\[3d_2 + 16 = 292\].
This step requires simplifying by first combining like terms, then subtracting 16 from both sides to isolate the term with \(d_2\). This results in the equation
\[3d_2 = 276\].
Solving for \(d_2\) is performed by dividing each side by 3, yielding
\[d_2 = 92\] miles.
After determining \(d_2\), we substitute back to discover \(d_1 = 70\) miles and \(d_3 = 130\) miles. This step-by-step equation solving showcases the logical process in tackling distance calculations, leading us to effectively classify the triangle.