If O(p/3, 4) is the midpoint of the line segment joining the points P(-6, 5) and Q(-2, 3), then what is the value of p?
Understand the Problem
The question states that point O(p/3, 4) is the midpoint of a line segment joining points P(-6, 5) and Q(-2, 3). We need to find the value of 'p'. We will use the midpoint formula to solve for p.
Answer
$p = -12$
Answer for screen readers
$p = -12$
Steps to Solve
- Apply the midpoint formula for the x-coordinate
The midpoint formula states that the midpoint between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. In this case, the x-coordinate of the midpoint O is $p/3$, and the x-coordinates of points P and Q are -6 and -2, respectively. Therefore, we can write the equation: $$ \frac{p}{3} = \frac{-6 + (-2)}{2} $$
- Simplify the right side of the equation
Simplify the expression on the right side of the equation: $$ \frac{p}{3} = \frac{-8}{2} $$ $$ \frac{p}{3} = -4 $$
- Solve for p
Multiply both sides of the equation by 3 to isolate p: $$ p = -4 \times 3 $$ $$ p = -12 $$
$p = -12$
More Information
The midpoint formula is a direct application of averaging the coordinates of two points. It is a fundamental tool in coordinate geometry.
Tips
A common mistake is to incorrectly apply the midpoint formula, such as adding instead of dividing by 2, or mixing up the x and y coordinates. Another mistake could be performing the arithmetic incorrectly when simplifying the equation.
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