Problem 44 - Precalculus Mathematics for Calculus

Question

Find the lengths of the medians of the triangle with vertices \(A(1,0), B(3,6),\) and \(C(8,2) .\) (A median is a line segment from a vertex to the midpoint of the opposite side.)

Step-by-Step Solution

Answer

The medians have lengths \( \frac{\sqrt{145}}{2} \), \( \sqrt{27.25} \), and \( \sqrt{37} \).

Step 1: Calculate Midpoint of Side BC

Identify the points B(3, 6) and C(8, 2). The midpoint, M, of line segment BC can be calculated as follows: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Therefore, \( M_{BC} = \left( \frac{3+8}{2}, \frac{6+2}{2} \right) = \left( \frac{11}{2}, 4 \right) \).

Step 2: Calculate the Length of Median AM

Find the length of the median from vertex A(1, 0) to the midpoint M of side BC found in Step 1. Apply the distance formula: \( AM = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Thus, \( AM = \sqrt{\left(\frac{11}{2} - 1\right)^2 + (4 - 0)^2} = \sqrt{\left(\frac{9}{2}\right)^2 + 16} = \sqrt{\frac{81}{4} + 16} = \sqrt{\frac{145}{4}} = \frac{\sqrt{145}}{2} \).

Step 3: Calculate Midpoint of Side AC

Identify the points A(1, 0) and C(8, 2). The midpoint, N, of AC is calculated as: \( N_{AC} = \left( \frac{1+8}{2}, \frac{0+2}{2} \right) = \left( 4.5, 1 \right) \).

Step 4: Calculate the Length of Median BN

Find the length of the median from vertex B(3, 6) to the midpoint N of side AC. The distance formula is applied: \( BN = \sqrt{(4.5 - 3)^2 + (1 - 6)^2} = \sqrt{(1.5)^2 + (-5)^2} = \sqrt{2.25 + 25} = \sqrt{27.25} \).

Step 5: Calculate Midpoint of Side AB

Identify the points A(1, 0) and B(3, 6). The midpoint, P, of AB is calculated as: \( P_{AB} = \left( \frac{1+3}{2}, \frac{0+6}{2} \right) = \left( 2, 3 \right) \).

Step 6: Calculate the Length of Median CP

Find the length of the median from vertex C(8, 2) to the midpoint P of side AB. Use the distance formula: \( CP = \sqrt{(2 - 8)^2 + (3 - 2)^2} = \sqrt{(-6)^2 + (1)^2} = \sqrt{36 + 1} = \sqrt{37} \).

Key Concepts

Midpoint Calculation

When dealing with triangle medians in geometry, one essential tool is the calculation of midpoints. The midpoint of a line segment is the point that is equidistant from both ends of the segment. It acts as a balancing point, neatly splitting the segment into two equal halves.
To find the midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

This calculation is simple and can be approached intuitively as finding the average of the x-coordinates and the y-coordinates.
For example, to find the midpoint of the line segment BC for vertices \(B(3, 6)\) and \(C(8, 2)\), we calculate:

• \( M = \left( \frac{3+8}{2}, \frac{6+2}{2} \right) = \left( \frac{11}{2}, 4 \right) \)

Thus, \(M_{BC}\) is \(\left( \frac{11}{2}, 4 \right)\), which supports us in pinpointing the medians' path in the triangle.

Distance Formula

Once we've established the midpoint, the next step in assessing triangle medians involves using the distance formula to find the length of median segments. The distance formula allows us to determine the straight-line distance between two points in a plane.
The formula is:

• \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

It is derived from the Pythagorean theorem and is used extensively in coordinate geometry to calculate distances between points. For instance, considering the median AM from vertex \(A(1, 0) \) to the midpoint \(M_{BC} \left( \frac{11}{2}, 4 \right)\), the computation uses:

• \( AM = \sqrt{\left( \frac{11}{2} - 1 \right)^2 + (4 - 0)^2} = \frac{\sqrt{145}}{2} \)

This calculation yields the length of the median AM, ensuring accuracy and precision in the geometric analysis.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, combines algebra and geometry using a coordinate system. It is essential in determining medians in triangles, particularly as it allows the use of algebraic techniques to solve geometric problems.
Every point in a plane can be depicted using coordinates \((x, y)\), which gives the precise location and straight-line connections between points on a plane. This geometric framework encompasses:

• Finding Points: Identify coordinates like A(1, 0), B(3, 6), and C(8, 2) for triangle vertices.
• Calculating Distances and Midpoints: Using the strategies highlighted in previous sections to understand triangle properties.

Coordinate geometry is transformative in geometry because it replaces abstract concepts with concrete calculations, turning geometry into a more accessible and solvable subject for students.