Question

In \(\triangle A B C, \overline{B Z}, \overline{C X}\), and \(\overline{A Y}\) are medians. If \(B Y=x-2\) and \(Y C=2 x-\) 10 , find the value of \(x\).

Short answer

The value of \(x\) is 8.

Step-by-step solution

Understand the Properties of a Median

In a triangle, a median is a line segment joining a vertex to the midpoint of the opposite side. For instance, in \(\triangle ABC\), \(\overline{AY}\) is a median, thus \(Y\) is the midpoint of \(\overline{BC}\). This means the segments \(\overline{BY}\) and \(\overline{YC}\) are equal.

Set Up an Equation

Since \(\overline{BY}\) and \(\overline{YC}\) are equal (as \(Y\) is the midpoint), we can equate their expressions: \(x-2 = 2x-10\).

Solve the Equation

Now solve the equation \(x-2 = 2x-10\). First, simplify the equation by subtracting \(x\) from both sides to get: \[-2 = x - 10\]. Next, add 10 to both sides to solve for \(x\): \[8 = x\].

Verify Your Solution

With \(x = 8\), check if the segments are indeed equal: calculate \(BY = 8 - 2 = 6\) and \(YC = 2(8) - 10 = 16 - 10 = 6\). Both are equal, confirming \(Y\) as the midpoint of \(\overline{BC}\).

Key concepts

Triangle

A triangle is a three-sided polygon, which is one of the most fundamental shapes in geometry. Every triangle consists of three edges and three vertices. Here are some simple facts about triangles that can help in solving geometric problems:

• The sum of the interior angles of a triangle is always 180 degrees.
• Triangles can be classified based on their sides as equilateral, isosceles, and scalene, or based on their angles as acute, right, or obtuse.
• In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side (Triangle Inequality Theorem).

Understanding these basic properties can greatly assist in solving problems related to triangles, such as finding medians, midpoints, or solving equations involving side lengths.

Median

In geometry, a median of a triangle is a line segment that joins one vertex to the midpoint of the opposite side. Each triangle has three medians, and these medians have some interesting properties:

• They always intersect at a single point called the centroid, which is the "center of mass" or "balance point" of the triangle.
• The centroid divides each median into two segments, with the segment closest to the vertex being twice as long as the segment closest to the midpoint of the side.
• The medians also divide the triangle into six smaller triangles of equal area.

Because medians are all about dividing the sides equally, they are crucial in problems that require understanding of balance and symmetry within triangles.

Midpoint

The midpoint is a very straightforward concept in geometry. It refers to the exact middle point of a line segment and can be found using the average of the endpoints.

• The coordinates of the midpoint of a line segment with endpoints egin{align*}(x_1, y_1) \text{ and } (x_2, y_2)egin{align*}are calculated using the formula: \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).
• In a triangle, a median is defined by its ability to connect a vertex with the midpoint of the opposite side, as noted in this exercise.
• Identifying the midpoint accurately is crucial because it often serves as a foundation for further geometrical construction and proof.

The concept of midpoint naturally leads us to understand the equality of divided segments, which is essential in an equation-solving context like in this problem.

Equation Solving

Equation solving is a process whereby we find the value of the unknown variable that makes an equation true. In geometry, this often involves setting up equations based on geometric properties or measurements.

• Start by writing the equation that reflects the given relationships or properties. In our case, it's equating the two expressions for parts of a median: \(x - 2 = 2x - 10\).
• Use basic algebraic operations to manipulate the equation. Here, the goal is to isolate the variable \(x\). This involves moving terms strategically, such as subtracting \(x\) from both sides to simplify.
• Perform arithmetic to solve the remaining simpler equation: add 10 to both sides, leading to \[-2 + 10 = x\], which simplifies to \[8 = x\].

Always verify your solution by plugging the value back into the original context to ensure it satisfies all conditions. This step reassures the accuracy and validity of your solution.