Finding Coordinates A And B Using The Midpoint Formula In Coordinate Geometry

In coordinate geometry, finding the midpoint of a line segment is a fundamental concept. The midpoint formula provides a straightforward method to calculate the coordinates of the midpoint given the coordinates of the endpoints. This article delves into a problem where we are given the coordinates of two points, P and Q, and the midpoint of the line segment PQ. Our goal is to determine the values of the unknowns, a and b, using the midpoint formula and algebraic techniques.

Problem Statement

Let's consider the problem at hand. We are given two points:

• Point P has coordinates (4a, 11b).
• Point Q has coordinates (-16, -5b).
• The midpoint of the line segment PQ has coordinates (6a, 33).

Our objective is to find the values of a and b. This problem combines the concepts of coordinate geometry and algebra, requiring us to apply the midpoint formula and solve a system of equations. Let's begin by understanding the midpoint formula and how it applies to this problem.

Understanding the Midpoint Formula

The midpoint formula is a crucial tool in coordinate geometry. It states that the coordinates of the midpoint M of a line segment with endpoints P(x₁, y₁) and Q(x₂, y₂) are given by:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

In simpler terms, the midpoint's x-coordinate is the average of the x-coordinates of the endpoints, and the midpoint's y-coordinate is the average of the y-coordinates of the endpoints. This formula is derived from the concept of finding the average position between two points in a coordinate plane. To effectively solve our problem, we will apply this formula to the given coordinates of points P, Q, and their midpoint.

Applying the midpoint formula to our problem, we can set up equations based on the given coordinates. The x-coordinate of the midpoint is the average of the x-coordinates of P and Q, and similarly for the y-coordinates. This will lead us to a system of equations that we can solve to find the values of a and b. Let's move on to setting up these equations and solving them systematically.

Setting Up the Equations

Now, let's apply the midpoint formula to the given points P(4a, 11b) and Q(-16, -5b) and the midpoint (6a, 33). According to the midpoint formula, the coordinates of the midpoint are calculated as follows:

• Midpoint x-coordinate: (4a + (-16))/2
• Midpoint y-coordinate: (11b + (-5b))/2

We are given that the midpoint has coordinates (6a, 33). Therefore, we can set up two equations by equating the calculated midpoint coordinates with the given midpoint coordinates:

1. (4a - 16)/2 = 6a
2. (11b - 5b)/2 = 33

These equations represent a system of two equations with two unknowns, a and b. Solving this system will give us the values of a and b that satisfy the given conditions. The first equation involves only a, while the second equation involves only b, making them relatively straightforward to solve. In the next section, we will proceed to solve these equations step by step.

Solving for A

Let's solve the first equation for a:

(4a - 16)/2 = 6a

To begin, we can multiply both sides of the equation by 2 to eliminate the fraction:

4a - 16 = 12a

Next, we want to isolate the terms containing a on one side of the equation. We can subtract 4a from both sides:

-16 = 12a - 4a

This simplifies to:

-16 = 8a

Now, to solve for a, we divide both sides by 8:

a = -16 / 8

Therefore:

a = -2*

We have successfully found the value of a. This was achieved by applying basic algebraic manipulations such as multiplying to eliminate fractions, isolating variables, and dividing to solve for the unknown. Now that we have the value of a, we can move on to solving the second equation for b. The process will be similar, involving algebraic steps to isolate b and find its value. Let's proceed to solving for b in the next section.

Solving for B

Now, let's solve the second equation for b:

(11b - 5b)/2 = 33

First, simplify the expression inside the parentheses:

(6b)/2 = 33

Now, simplify the fraction:

3b = 33

To solve for b, divide both sides of the equation by 3:

b = 33 / 3

Therefore:

b = 11*

We have successfully found the value of b. This involved simplifying the equation and then isolating b by dividing both sides by its coefficient. With the values of both a and b now determined, we can verify our solution by substituting these values back into the original equations. This will ensure that our solution is correct and that the midpoint coordinates align with the given information. Let's proceed to verify our solution in the next section.

Verifying the Solution

To verify our solution, we will substitute the values a = -2 and b = 11 back into the original coordinates of points P and Q and the midpoint formula. This will ensure that our calculated values for a and b are correct and consistent with the given information.

First, let's find the coordinates of point P using a = -2 and b = 11:

P = (4a, 11b) = (4*(-2), 11*(11)) = (-8, 121)

Next, let's find the coordinates of point Q:

Q = (-16, -5b) = (-16, -5*(11)) = (-16, -55)

Now, let's use the midpoint formula to find the midpoint of PQ using our calculated coordinates:

Midpoint = ((-8 + (-16))/2, (121 + (-55))/2) = (-24/2, 66/2) = (-12, 33)

Finally, let's check if these midpoint coordinates match the given midpoint coordinates (6a, 33). Substituting a = -2, we get:

Given Midpoint x-coordinate = 6a = 6*(-2) = -12

The calculated midpoint (-12, 33) matches the midpoint we found using the midpoint formula. Therefore, our solution a = -2 and b = 11 is correct.

In this section, we verified our solution by substituting the values of a and b back into the original coordinates and applying the midpoint formula. The calculated midpoint coordinates matched the given midpoint coordinates, confirming the correctness of our solution. This verification step is crucial in problem-solving to ensure accuracy and confidence in the final answer. Now that we have verified our solution, let's summarize our findings and conclude the article.

Conclusion

In this article, we successfully determined the values of a and b given the coordinates of points P and Q and the midpoint of the line segment PQ. We were given:

• Point P (4a, 11b)
• Point Q (-16, -5b)
• Midpoint (6a, 33)

We utilized the midpoint formula to set up a system of equations and solved for a and b. Our calculations revealed that:

• a = -2*
• b = 11*

We then verified our solution by substituting these values back into the original coordinates and confirming that the calculated midpoint matched the given midpoint. This problem demonstrated the application of coordinate geometry principles and algebraic techniques to solve for unknowns. Understanding and applying the midpoint formula is crucial for solving various problems related to line segments and their midpoints in coordinate geometry. This article provides a comprehensive guide to solving such problems, from setting up the equations to verifying the solution. The skills and concepts discussed here are essential for further exploration of geometry and algebra.