Problem 5 - The Calculus with Analytic Geometry

Question

Find the length of the medians of the triangle having vertices \(A(2,3), B(3,-3)\), and \(C(-1,-1)\).

Step-by-Step Solution

Answer

The medians are approximately 5.1, 4.7, and 3.6 units long.

Step 1 - Find the Midpoints of the Sides

Calculate the midpoints of each side of the triangle.Midpoint of AB is \(\frac{2+3}{2}, \frac{3+(-3)}{2}\), which simplifies to (2.5, 0)\.Midpoint of BC is \(\frac{3+(-1)}{2}, \frac{-3+(-1)}{2}\), which simplifies to (1, -2)\.Midpoint of CA is \(\frac{-1+2}{2}, \frac{-1+3}{2}\), which simplifies to (0.5, 1)\.

Step 2 - Use the Distance Formula to Find the Length of the Medians

The length of the median can be found using the distance formula \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)1. Median from A to BC: \(\text{Distance} = \sqrt{(2-1)^2 + (3 - (-2))^2}\ = \sqrt{1 + 25}\ = \sqrt{26}\) or approximately 5.1 units.2. Median from B to CA: \(\text{Distance} = \sqrt{(3-0.5)^2 + (-3-1)^2}\ = \sqrt{6.25 + 16}\ = \sqrt{22.25}\) or approximately 4.7 units.3. Median from C to AB: \(\text{Distance} = \sqrt{(-1-2.5)^2 + (-1-0)^2}\ = \sqrt{12.25 + 1}\ = \sqrt{13.25}\) or approximately 3.6 units.

Key Concepts

Midpoints of Sides

In a triangle, the midpoint of a side is the point that is equidistant from both endpoints of that side. This means it is exactly halfway along the segment. To calculate the midpoint of a line segment when you know the coordinates of its endpoints, you use the midpoint formula. The formula is \(\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Let's apply this to our specific example.

For the triangle with vertices at \(A(2,3)\), \(B(3,-3)\), and \(C(-1,-1)\):

• Midpoint of \(AB\) is calculated as \(\left(\frac{2+3}{2}, \frac{3+(-3)}{2}\right) = (2.5, 0)\).
• Midpoint of \(BC\) is \(\left(\frac{3+(-1)}{2}, \frac{-3+(-1)}{2}\right) = (1, -2)\).
• Midpoint of \(CA\) is \(\left(\frac{-1+2}{2}, \frac{-1+3}{2}\right) = (0.5, 1)\).

Calculating midpoints is a crucial first step in many geometry problems, especially when dealing with medians.

Distance Formula

The distance formula helps to find the length between two points in a coordinate plane. This formula is derived from the Pythagorean theorem. The distance \(d\) between points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula takes into account the horizontal and vertical distances between the points to give the straight-line distance, or hypotenuse, connecting them.
In our exercise, we use the distance formula to find the lengths of the medians from each vertex to the midpoint of the opposite side. Let's look at each median:

• Median from \(A\) to midpoint of \(BC\): \( \text{Distance} = \sqrt{(2-1)^2 + (3 - (-2))^2} = \sqrt{1 + 25} = \sqrt{26}\) or approximately 5.1 units.
• Median from \(B\) to midpoint of \(CA\): \( \text{Distance} = \sqrt{(3-0.5)^2 + (-3-1)^2} = \sqrt{6.25 + 16} = \sqrt{22.25} \) or approximately 4.7 units.
• Median from \(C\) to midpoint of \(AB\): \( \text{Distance} = \sqrt{(-1-2.5)^2 + (-1-0)^2} = \sqrt{12.25 + 1} = \sqrt{13.25} \) or approximately 3.6 units.

Length of Medians

A median in a triangle is a line segment joining a vertex to the midpoint of the opposite side. In essence, it splits the triangle into two smaller triangles of equal area. Calculating the length of medians can be done through the distance formula once the midpoints are known.
For a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), and midpoints as calculated previously, the lengths of the medians were found as follows:

• Median from \(A\) to midpoint of \(BC\) is roughly 5.1 units.
• Median from \(B\) to midpoint of \(CA\) is approximately 4.7 units.
• Median from \(C\) to midpoint of \(AB\) is about 3.6 units.

Understanding these calculations is vital for further explorations in triangle properties and proofs. These principles can be applied to various other problems and higher-level math concepts.