The coordinates of the midpoint of the line segment joining  A(2p+1,4) B(5,q-1) are (2p,q). Find the value of p and q

To solve for the values of \( p \) and \( q \), let's use the midpoint formula.

The formula for the midpoint \( M(x, y) \) of a line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:

\[
M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Given:
- Point \( A(2p + 1, 4) \)
- Point \( B(5, q - 1) \)
- Midpoint \( M(2p, q) \)

We can apply the midpoint formula to the given points:

\[
\left(\frac{(2p + 1) + 5}{2}, \frac{4 + (q - 1)}{2}\right) = (2p, q)
\]

This gives us two equations by equating the \( x \) and \( y \) coordinates:

1. For the \( x \)-coordinates:
\[
\frac{(2p + 1) + 5}{2} = 2p
\]

2. For the \( y \)-coordinates:
\[
\frac{4 + (q - 1)}{2} = q
\]

Now, let's solve these equations one by one.

### Equation 1: Solving for \( p \)
\[
\frac{2p + 1 + 5}{2} = 2p
\]

\[
\frac{2p + 6}{2} = 2p
\]

\[
p + 3 = 2p
\]

\[
3 = 2p - p
\]

\[
p = 3
\]

### Equation 2: Solving for \( q \)
\[
\frac{4 + q - 1}{2} = q
\]

\[
\frac{q + 3}{2} = q
\]

\[
q + 3 = 2q
\]

\[
3 = 2q - q
\]

\[
q = 3
\]

### Final Answer:
The values are \( p = 3 \) and \( q = 3 \).

Would you like any more details, or do you have any other questions?

Here are some related questions you might find interesting:
1. How do you derive the midpoint formula?
2. What are some applications of the midpoint in geometry?
3. How do you find the length of the line segment between two points?
4. What is the distance formula in coordinate geometry?
5. How do you find the equation of a line given two points?

**Tip:** When dealing with coordinates, double-check the formulae before substituting values to avoid calculation errors.

Learn how to find the values of p and q using the midpoint formula in coordinate geometry. Detailed step-by-step solution provided for the problem involving points A(2p+1, 4) and B(5, q-1) with midpoint (2p, q). Suitable for high school students.

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