Problem 50 - Algebra and Trigonometry

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

\(\frac{\sqrt{145}}{2}, \frac{\sqrt{109}}{2}, \sqrt{37}\)

Step 1: Calculate Midpoint of Side BC

To find the median from vertex A, we first need the midpoint of side BC. The midpoint formula is \( \text{Midpoint} = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). Use the coordinates of B and C: \((3,6)\) and \((8,2)\). The midpoint is \( \left( \frac{3+8}{2}, \frac{6+2}{2} \right) = \left( \frac{11}{2}, 4 \right) \).

Step 2: Find Length of Median from A to BC

Now, use the distance formula to find the length of the median from A to the midpoint of BC. The distance formula is \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substituting in the coordinates of A \((1,0)\) and the midpoint \( \left( \frac{11}{2}, 4 \right) \), we get \( d = \sqrt{\left(\frac{11}{2} - 1\right)^2 + (4 - 0)^2} = \sqrt{\left(\frac{9}{2}\right)^2 + 16} = \sqrt{\frac{81}{4} + 16} = \sqrt{\frac{145}{4}} = \frac{\sqrt{145}}{2} \).

Step 3: Calculate Midpoint of Side AC

Next, calculate the midpoint of side AC to find the median from vertex B. The coordinates of A and C are \((1,0)\) and \((8,2)\). The midpoint of AC is \( \left( \frac{1+8}{2}, \frac{0+2}{2} \right) = (4.5, 1) \).

Step 4: Find Length of Median from B to AC

Using the distance formula from vertex B to the midpoint of AC, i.e.,\( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substitute the coordinates of B \((3,6)\) and midpoint \((4.5, 1)\): \( d = \sqrt{(4.5 - 3)^2 + (1 - 6)^2} = \sqrt{1.5^2 + (-5)^2} = \sqrt{2.25 + 25} = \sqrt{27.25} = \frac{\sqrt{109}}{2} \).

Step 5: Calculate Midpoint of Side AB

Calculate the midpoint of side AB for the median from vertex C. The coordinates of A and B are \((1,0)\) and \((3,6)\). The midpoint is \( \left( \frac{1+3}{2}, \frac{0+6}{2} \right) = (2, 3) \).

Step 6: Find Length of Median from C to AB

Find the length of the median from C to the midpoint of AB using the distance formula: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substitute coordinates of C \((8,2)\) and midpoint \((2,3)\): \( d = \sqrt{(2 - 8)^2 + (3 - 2)^2} = \sqrt{(-6)^2 + 1} = \sqrt{36 + 1} = \sqrt{37} \).

Key Concepts

Midpoint Formula

In geometry, the midpoint formula is an essential tool for determining the middle point of a line segment connecting two points in a coordinate plane. This is particularly useful when finding the median in a triangle, as it involves identifying the midpoint of the opposite side from the vertex.

To calculate a midpoint, you use the formula:

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

For example, if you have points B(3, 6) and C(8, 2), their midpoint M of BC is calculated as follows:

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

This means the midpoint M of side BC is \( \left( \frac{11}{2}, 4 \right) \). Finding this midpoint is the first step in calculating the length of a median from vertex A to side BC, connecting A directly to this midpoint. By mastering the midpoint formula, you can easily tackle similar problems involving medians in triangles.

Distance Formula

The distance formula is a powerful method used to calculate the length between two points in a coordinate plane. When dealing with medians in triangles, this formula helps determine the length of the median itself.

To apply the distance formula, use:

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

For example, to find the length of the median from A to the midpoint of BC, given A(1, 0) and M(\( \frac{11}{2}, 4 \)), substitute the coordinates into the formula:

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

This formula is consistently reliable for such calculations. By understanding its use, you can find exact measurements between points, a necessary skill in many geometric problems.

Median Length Calculation

Calculating the median length in a triangle involves combining both the midpoint and the distance formula. The median is a line from a vertex to the midpoint of the opposite side, and its length gives insight into the triangle's dimensions.

Let's process step by step the calculation for one median:

• First, determine the midpoint of side BC, which is \( \left( \frac{11}{2}, 4 \right) \) as calculated using the midpoint formula.
• Next, apply the distance formula using point A(1, 0) and the midpoint: \( \frac{\sqrt{145}}{2} \).

Similarly, these steps are repeated for the other medians from vertices B and C utilizing their respective midpoints of opposite sides:

• Median from B to midpoint of AC: \( \frac{\sqrt{109}}{2} \)
• Median from C to midpoint of AB: \( \sqrt{37} \)

Through these calculations, each median gives a different insight into the structure of the triangle, which aids in understanding its overall geometry.