Question

In \(\triangle A B C\), the coordinates of the vertices are \(A(2,4)\), \(B(-3,6)\), and \(C(-5,-2)\). To the nearest tenth, what is the measure of the median drawn from \(A\) to \(\overline{B C}\) ? Include a drawing on a coordinate plane of the triangle and the median.

Short answer

The measure of the median is approximately 6.3 units.

Step-by-step solution

Calculate the midpoint of BC

The median from vertex \(A\) to \(\overline{BC}\) passes through the midpoint of \(\overline{BC}\). To find this midpoint \((M)\), use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] where \((x_1, y_1) = B(-3, 6)\) and \((x_2, y_2) = C(-5, -2)\). Plugging in these coordinates gives: \[ M = \left( \frac{-3 + (-5)}{2}, \frac{6 + (-2)}{2} \right) = \left( \frac{-8}{2}, \frac{4}{2} \right) = (-4, 2) \].

Calculate the length of the median

The length of the median from \(A(2,4)\) to \(M(-4,2)\) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substitute \((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (-4, 2)\): \[ d = \sqrt{((-4) - 2)^2 + (2 - 4)^2} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \] The distance \(\sqrt{40}\) can be simplified to \(2\sqrt{10}\), which is approximately \(6.3\) when rounded to the nearest tenth.

Visualize with a drawing

To fully understand the problem, plot \(A(2, 4)\), \(B(-3, 6)\), \(C(-5, -2)\) and \(M(-4, 2)\) on a coordinate plane. Draw the triangle \(\triangle ABC\) and the median \(AM\) from \(A\) to midpoint \(M\) of \(BC\). This helps in visualizing how the median divides the triangle into two smaller triangles of equal area.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a crucial branch of mathematics. It merges algebra and geometry using a coordinate system to describe geometric shapes. This system allows us to represent points, lines, and shapes as equations and perform calculations to understand their properties better.
In this particular problem, we are working with triangle geometry on a coordinate plane. The vertices of the triangle are given specific coordinates. This lets us apply various formulas to find lengths, midpoints, and other valuable geometric properties. By plotting these points on a coordinate plane, you get a visual representation that provides further insight into the problem.

Midpoint Formula

The midpoint formula helps find the central point between two defined coordinates in a 2D plane. This is key when dealing with geometry problems like finding the median of a triangle.
Given two points,

• Point 1: \((x_1, y_1)\)
• Point 2: \((x_2, y_2)\)

The formula for finding the midpoint \(M\) is:

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

In our example, the midpoint between vertices \(B(-3, 6)\) and \(C(-5, -2)\) was calculated as \((-4, 2)\). This point serves as a target through which the median from vertex A will pass.

Distance Formula

The distance formula is another fundamental tool in coordinate geometry. It allows us to calculate the distance between two points on a plane. This is essential for finding the actual length of lines in geometric shapes.
The formula for the distance \(d\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

In our solved problem, we used this formula to calculate the length of the median from point \(A(2, 4)\) to the midpoint \(M(-4, 2)\), yielding the distance \(\sqrt{40}\), which approximates to 6.3 when rounded.

Median of a Triangle

A median of a triangle is a line segment that stretches from a vertex to the midpoint of the opposite side. It splits the triangle into two smaller triangles of equal area. Medians have interesting properties and play a vital role in geometry.
The problem illustrates finding the median from vertex \(A\) to side \(\overline{BC}\). By calculating the midpoint of \(BC\) and using the distance formula from \(A\), we can determine the median's length, allowing us to appreciate the geometric relationships present in the triangle. Understanding these concepts enhances our ability to visualize and solve more complex geometric problems in coordinate geometry.