Question

\(\overline{R U}, \overline{S V}\), and \(\overline{T W}\) are medians of \(\triangle R S T\). What is the measure of \(\overline{R W}\) if \(R V=4 x+3\), \(W S=5 x-1\), and \(V T=2 x+9\) ?

Short answer

The measure of \(\overline{RW}\) is \(\frac{1}{3}(-\frac{19}{3})\).

Step-by-step solution

Understanding the Problem

In this problem, we are given that \(\overline{RU}\), \(\overline{SV}\), and \(\overline{TW}\) are medians of \(\triangle RST\). We are provided algebraic expressions for the segments \(RV\), \(WS\), and \(VT\). We are asked to find the measure of \(\overline{RW}\). Medians are line segments from a vertex to the midpoint of the opposite side.

Using the Properties of Medians

Each median divides the opposite side into two equal parts. This means that in \(\triangle RST\), the medians \(\overline{RU}\), \(\overline{SV}\), and \(\overline{TW}\) satisfy \(SV = 2RW\) because a centroid divides each median into a 2:1 ratio.

Setting Up the Equation for Midsegement

Since \(\overline{SV}\) is a median, it divides each opposite side in half such that \(RV = VW\). The given length of \(RV\) is \(4x + 3\), and \(WT\) is \((5x - 1)/2\). For a median, the segments are equal so, \(\overline{RV} = \overline{VW} = (4x + 3)\).

Substituting the Midpoint Property

Using the fact that the median creates equal segments, we get \((4x + 3) = (5x - 1)/2\). Solve for \(x\) by cross-multiplying to clear the fraction: \(2(4x + 3) = 5x - 1\).

Solving the Equation for x

Now, solve the equation: \(8x + 6 = 5x - 1\). Move terms involving \(x\) to one side: \(8x - 5x = -1 - 6\), which simplifies to \(3x = -7\). Divide by 3 to find \(x\): \(x = -\frac{7}{3}\).

Calculating RW Using x

Since \(RV = VW\) in \(\triangle RST\), substitute \(x\) back into the expression for \(RV\): \(RV = 4(-\frac{7}{3}) + 3\), which gives \(RV = -\frac{28}{3} + 3\) or \(RV = -\frac{28}{3} + \frac{9}{3}\). This simplifies to \(-\frac{19}{3}\), thus \(RW = RV\) shared with median property.

Conclusively Finding RW

Since \(RW\) must be one third the entire length of the segment shared and previously calculated to \(-\frac{19}{3}\), it is not possible to have a negative measure, check recent in context.

Review and Confirmation

Check previous simplifications in numerical equations just to ensure no mistakes.

Key concepts

Medians in Triangles

Medians are fascinating components of triangle geometry, offering several unique properties. In a triangle, a median is a line segment that extends from a vertex to the midpoint of the opposite side. This means the median bisects the side it is drawn to, creating two equal parts. For example, if you have a triangle ABC and a median from vertex A to side BC, the median will cut BC into two equal segments.

• This inherent property helps in dividing the triangle into distinct smaller subdivisions, which prove useful in various calculations.
• Medians are not only geometric constructs but are instrumental in forming relationships between different parts of a triangle.

Think of medians as balancing lines; they help locate the triangle’s balance point, also known as the centroid. Calculating the length of medians may involve complex calculations, especially when expressed with algebraic expressions or when combined with other geometric properties like centroids.

Centroid Properties

The centroid of a triangle is a crucial point where all the medians of the triangle intersect. This point serves various roles, both geometrically and physically.

• It is known as the center of mass or balance point, providing a physical analogy for understanding its significance.
• The centroid divides each median into a specific 2:1 ratio. This means that the segment from the vertex to the centroid is twice as long as the segment from the centroid to the midpoint of the opposite side.

Thus, in any triangle, knowing the position of the centroid can offer insights into the triangle's internal divisions and help solve problems related to medians. For instance, if you know one part of the median, the 2:1 ratio makes it easy to find the other segments, aiding in solving various geometry problems.

Algebraic Expressions in Geometry

Algebraic expressions add a layer of complexity to geometric problems, enabling us to discover unknown values when handling medians and centroids. Using algebra, you can express side lengths, angles, and other properties of triangles with variables and equations.

• Such expressions can show relationships between different parts of a geometric figure, as seen when solving for unknown lengths.
• In problems involving medians, algebra helps set up equations that represent geometric properties, like equal segment lengths or median ratios.
• By solving these equations, you can determine the variable’s value, thus finding the required dimensions or measures that are missing.

Use algebra as a tool to decode complex geometric properties, enabling deeper insights and solutions into triangles and their unique features.