Question

The equation of median drawn to the side \(B C\) of \(\triangle A B C\) whose verticles are \(A(1,-2), B(3,6)\) and \(C(5,0)\) is (1) \(5 x-3 y-11=0\) (2) \(5 x+3 y-11=0\) (3) \(3 x-5 y+11=0\) (4) \(3 x-5 y-11=0\)

Short answer

Answer: The equation of the median is (1) \(5x - 3y - 11 = 0\).

Step-by-step solution

Find the coordinates of the midpoint of BC

First, we need to find the midpoint of BC. We can use the midpoint formula, which is \(M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\). So, the coordinates of the midpoint of BC are: $$M = \left(\frac{3+5}{2}, \frac{6+0}{2}\right) = \left(4, 3\right)$$

Calculate the slope of the median

The slope of the median can be calculated using the slope formula, \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, we need the slope of the line passing through points A(1, -2) and M(4, 3). Calculating the slope, we get: $$m = \frac{3 - (-2)}{4 - 1} = \frac{5}{3}$$

Use the point-slope form to find the equation of the median

Now that we have the slope of the median and a point through which it passes (A(1, -2)), we can use the point-slope form to find the equation of the median. The point-slope form is given by \((y - y_1) = m(x - x_1)\). Plugging in the values, we get: $$(y + 2) = \frac{5}{3}(x - 1)$$

Simplify the equation and match it to the given options

To match our equation to given options, we need to eliminate the fractions and put our equation in the standard form. First, we'll multiply both sides by 3 to get rid of the fraction: $$3(y + 2) = 5(x - 1)$$ Now, we have: $$3y + 6 = 5x - 5 \Rightarrow 3y + 11 = 5x$$ which can be rewritten as: $$5x - 3y - 11 = 0$$ Comparing our equation with the given options, the correct answer is (1) \(5x - 3y - 11 = 0\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful tool for analyzing and understanding the relationships between points, lines, and shapes on a plane. In this method, each point is identified by a pair of numerical coordinates, usually represented as \(x, y\). These coordinates correspond to numerical values on the horizontal x-axis and vertical y-axis, respectively.
Through coordinate geometry, you can solve various mathematical problems by using methods such as equations and formulas. With the ability to measure distances, find slopes, and assess symmetrical properties, coordinate geometry facilitates a clearer understanding of geometric relationships.
In the case of triangles, such as \(\triangle ABC\) in our exercise, understanding the coordinates of its vertices can help you find specific lines and points using geometric calculations, like medians or midpoints.

Midpoint Formula

The midpoint formula is a crucial component in coordinate geometry, making it easier to determine the exact middle point between two given points in a plane. For points \(B(x_1, y_1)\) and \(C(x_2, y_2)\), the formula to find their midpoint \(M(x, y)\) is:

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

This formula essentially averages the x-coordinates and the y-coordinates of \(B\) and \(C\), providing a new point \(M\) that lies exactly in the middle.
In our exercise, we applied the midpoint formula to \(B(3, 6)\) and \(C(5, 0)\), which gave us \(M(4, 3)\) as the midpoint of side \(BC\) in \(\triangle ABC\). Discovering the midpoint is a key step in constructing the median from \(A\) to \(BC\).

Slope of a Line

The slope of a line describes how steep the line is and its direction. It is an essential measure in coordinate geometry because it gives insight into the gradient of the line between two points \(A(x_1, y_1)\) and \(M(x_2, y_2)\). The formula for finding the slope \(m\) is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula calculates the 'rise over run,' which is the change in the y-coordinates over the change in the x-coordinates.
In the exercise, we found the slope of the median line connecting \(A(1, -2)\) and \(M(4, 3)\). Substituting these coordinates into the slope formula, we calculated a slope of \(\frac{5}{3}\). Knowing this slope is necessary to determine the line's equation accurately using the point-slope form.

Point-Slope Form

The point-slope form of a line is a formula used to find the equation of a line when you know the slope of the line and a specific point it passes through. This form is expressed as:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) is the slope, and \(x_1, y_1\) are the coordinates of a known point on the line.
In our exercise, we used this form to find the median's equation in \(\triangle ABC\) passing through point \(A(1, -2)\) with the slope \(\frac{5}{3}\). Applying the values, we have:
\(y + 2 = \frac{5}{3}(x - 1)\).
Converting this into a usable equation form helps us match it to any given options. Simplification eventually led us to the form \(5x - 3y - 11 = 0\), matching the provided answer choice (1).