Problem 44 - 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

Medians are \(\frac{\sqrt{145}}{2}\), \(\frac{\sqrt{109}}{2}\), and \(\sqrt{37}\).

Step 1: Find Midpoints of Each Side

To find the median of a triangle, begin by determining the midpoint of each side. We use the midpoint formula for a line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\): \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\). - For side \(BC\), with endpoints \(B(3,6)\) and \(C(8,2)\), the midpoint is: \[ \left( \frac{3+8}{2}, \frac{6+2}{2} \right) = \left(\frac{11}{2}, 4\right) \]- For side \(AC\), with endpoints \(A(1,0)\) and \(C(8,2)\), the midpoint is: \[ \left( \frac{1+8}{2}, \frac{0+2}{2} \right) = \left(\frac{9}{2}, 1\right) \] - For side \(AB\), with endpoints \(A(1,0)\) and \(B(3,6)\), the midpoint is: \[ \left( \frac{1+3}{2}, \frac{0+6}{2} \right) = (2, 3) \]

Step 2: Use Distance Formula to Find Medians

Now, use the distance formula to find each median, given by two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]- Median from \(A\) to midpoint of \(BC\): \[ d = \sqrt{\left(\frac{11}{2} - 1\right)^2 + \left(4 - 0\right)^2} = \sqrt{\left(\frac{9}{2}\right)^2 + 4^2} = \sqrt{\frac{81}{4} + 16} = \sqrt{\frac{145}{4}} = \frac{\sqrt{145}}{2} \]- Median from \(B\) to midpoint of \(AC\): \[ d = \sqrt{\left(\frac{9}{2} - 3\right)^2 + \left(1 - 6\right)^2} = \sqrt{\left(\frac{3}{2}\right)^2 + (-5)^2} = \sqrt{\frac{9}{4} + 25} = \sqrt{\frac{109}{4}} = \frac{\sqrt{109}}{2} \]- Median from \(C\) to midpoint of \(AB\): \[ d = \sqrt{\left(2 - 8\right)^2 + \left(3 - 2\right)^2} = \sqrt{(-6)^2 + 1^2} = \sqrt{36 + 1} = \sqrt{37} \]

Step 3: Conclusion

The lengths of the medians of the triangle are: - Median from vertex \(A\) to \(BC\): \(\frac{\sqrt{145}}{2}\)- Median from vertex \(B\) to \(AC\): \(\frac{\sqrt{109}}{2}\)- Median from vertex \(C\) to \(AB\): \(\sqrt{37}\)

Key Concepts

Midpoint Formula

The midpoint formula is a crucial tool in geometry, particularly when dealing with triangles and their medians. A median of a triangle connects a vertex to the midpoint of the opposite side. To locate this midpoint, we use the midpoint formula. If you have endpoints of a line segment as

• \((x_1, y_1)\)
• \((x_2, y_2)\)

The midpoint is given by:

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

Using this formula, you can determine the midpoint of any side of the triangle. This is the first step towards finding the medians of the triangle.

Distance Formula

Once you have the midpoints, the distance formula helps you find the length of the medians. The distance formula calculates the length of a line segment that joins two points. If the points are

• \((x_1, y_1)\)
• \((x_2, y_2)\)

The formula to find the distance \(d\) is:

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

By applying this formula to the vertex of a triangle and the midpoint of the opposite side, you can compute the lengths of the medians, ensuring accuracy and understanding of spatial relationships.

Geometry

Geometry, as a branch of mathematics, helps us understand shapes, sizes, and the properties of space. In the context of triangles, understanding how to find the lengths of medians is an essential skill.
Here are some fundamental concepts in geometry that relate to medians:

• Triangles have three sides and three vertices.
• The median divides the triangle into two smaller triangles of equal area.
• A triangle's centroid is where its three medians intersect, offering unique properties, such as balancing the triangle perfectly.

A strong grasp of geometry aids in visualizing and calculating precise outcomes, enhancing your problem-solving skills.

Triangle Vertices

Vertices are the corner points of a triangle. Each triangle has three vertices, and they are crucial in defining the shape and size of the triangle.
In a coordinate plane:

• Vertices are represented as ordered pairs \((x, y)\).
• The location of each vertex is crucial in calculating distances and midpoints.
• Understanding vertices allows you to apply geometric formulas accurately.

By knowing the coordinates of all three vertices, calculations such as finding midpoints and distances become straightforward, forming the base for more complex geometric analysis.

Triangle Medians

A median in a triangle is a line segment from a vertex to the midpoint of the opposite side. It's a vital element in understanding the balance and symmetry of triangles. Here's why medians matter:

• There are three medians in a triangle, and they all converge at a point known as the centroid.
• The centroid is interesting because it is always two-thirds of the distance from each vertex along a median.
• Finding medians involves using the midpoint and distance formulas, thus linking algebra and geometry.

Grasping the concept of triangle medians not only enhances spatial awareness but also provides insights into a triangle's properties and behaviors.