Problem 2 - 5 Steps to A 5: AP Calculus AB 2021

Question

The vertices of a triangle are \(A(-2,0), B(0,6),\) and \(C(4,0) .\) Find an equation of a line containing the median from vertex \(A\) to \(\overline{B C}\).

Step-by-Step Solution

Answer

Answer: The equation of the line containing the median from vertex A to the midpoint of side BC is \(y = \frac{3}{4}x + \frac{3}{2}\).

Step 1: Calculate the midpoint (D) of side BC

We use the midpoint formula: \(D (x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) Using points B\((0, 6)\) and C\((4, 0)\), we find point D: \(D \left( \frac{0 + 4}{2}, \frac{6 + 0}{2} \right) = \left( 2, 3 \right)\)

Step 2: Calculate the slope of the line passing through points A and D

The slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Using points A\((-2, 0)\) and D\((2, 3)\), we find the slope: \(m = \frac{3 - 0}{2 - (-2)} = \frac{3}{4}\)

Step 3: Find the equation of the line using point-slope form

Let's consider point A\((-2, 0)\) and the calculated slope \(m = \frac{3}{4}\). The point-slope equation of the line: \(y - y_1 = m (x - x_1)\) By plugging in the values, you will get: \(y - 0 = \frac{3}{4}(x - (-2))\)

Step 4: Convert point-slope form to slope-intercept form

Simplify the equation in step 3 to convert it to the slope-intercept form (y = mx + b): \(y = \frac{3}{4}(x + 2)\) which is equivalent to: \(y = \frac{3}{4}x + \frac{3}{2}\) The equation of the line containing the median from vertex A to side BC is: \(\boxed{y = \frac{3}{4}x + \frac{3}{2}}\).

Key Concepts

Midpoint Formula

The midpoint formula is a tool that helps us find the center point of a line segment. If you have two points,

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

the midpoint \(D(x, y)\) is calculated as follows:\[D(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula essentially averages the x-coordinates and the y-coordinates of the two endpoints.

In our exercise, we used this formula to find the midpoint of side BC in a triangle with vertices at B(0, 6) and C(4, 0).By inserting these values into the midpoint formula, we find:\[D = \left( \frac{0 + 4}{2}, \frac{6 + 0}{2} \right) = (2, 3)\]This midpoint D is an important point because it serves as one endpoint of the median line from vertex A.

Slope Formula

The slope of a line is an indicator of its steepness and direction. It is calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \((x_1, y_1)\) is the first point
• \((x_2, y_2)\) is the second point on the line

The value of the slope tells us how much the line rises or falls as we move along the x-axis. A positive slope means the line rises, while a negative slope means it falls.

In the exercise, we calculated the slope of the line connecting point A(-2, 0) and the midpoint D(2, 3) using this formula:\[m = \frac{3 - 0}{2 - (-2)} = \frac{3}{4}\]This means for every 4 units you move to the right along the x-axis, the line ascends 3 units. Understanding slopes in this way is crucial for analyzing and graphing lines accurately.

Point-Slope Form

The point-slope form of a line's equation is useful when you know one point on the line and its slope. It is represented as:\[y - y_1 = m(x - x_1)\]where

• \((x_1, y_1)\) is a known point on the line
• \(m\) is the slope

This form allows you to quickly write the equation of a line when these two values are available.

In the exercise, we used point A(-2, 0) and the slope \(m = \frac{3}{4}\) to form the equation:\[y - 0 = \frac{3}{4}(x - (-2))\]Simplifying helps us transition this into a more familiar and easy-to-use format, which we'll discuss next.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is presented as:\[y = mx + b\]where

• \(m\) is the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis

This form provides a clear view of the line's slope and starting point, making it easy to graph and understand the line's behavior.

In the exercise, by transforming the point-slope form equation, we arrived at:\[y = \frac{3}{4}x + \frac{3}{2}\]This tells us that the line rises \(3\) units for every \(4\) units it moves to the right and crosses the y-axis at \((0, \frac{3}{2})\). Understanding this form is essential for quickly sketching out or interpreting the line on a graph.