If B is the midpoint of AC and AB = 2x+5 and BC = 4x-9 Find x, AB,BC,AC

Given the problem:

- **B is the midpoint of AC**.
- **AB = 2x + 5**.
- **BC = 4x - 9**.

Since B is the midpoint of AC, we know that:

\[ AB = BC \]

So, we can set the expressions for AB and BC equal to each other:

\[
2x + 5 = 4x - 9
\]

### Step 1: Solve for \( x \)

First, subtract \( 2x \) from both sides:

\[
5 = 2x - 9
\]

Next, add 9 to both sides:

\[
14 = 2x
\]

Finally, divide both sides by 2:

\[
x = 7
\]

### Step 2: Find the lengths of AB, BC, and AC

Now that we have \( x = 7 \), we can substitute this value back into the expressions for AB and BC:

- **AB = 2x + 5**
  
  \[
  AB = 2(7) + 5 = 14 + 5 = 19
  \]

- **BC = 4x - 9**
  
  \[
  BC = 4(7) - 9 = 28 - 9 = 19
  \]

Since B is the midpoint of AC, we know that:

\[
AC = AB + BC
\]

So,

\[
AC = 19 + 19 = 38
\]

### Final Answer:
- \( x = 7 \)
- \( AB = 19 \)
- \( BC = 19 \)
- \( AC = 38 \)

Would you like further details or have any questions?

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Here are 8 related questions you might find interesting:
1. What is the definition of a midpoint in geometry?
2. How do you find the midpoint of a line segment given two points?
3. If \( AB = 3x + 2 \) and \( BC = 5x - 6 \), how would you find x when B is the midpoint?
4. How can you verify that B is the midpoint using coordinate geometry?
5. If \( AB = 4y + 7 \) and \( BC = 6y - 5 \), and B is the midpoint, what are the values of y and AC?
6. How do you use the distance formula in problems involving midpoints?
7. What happens to the length of AB and BC if B is no longer the midpoint of AC?
8. Can the value of x be negative in such midpoint problems?

**Tip:** Always check your final answers by substituting the values back into the original equations to ensure they satisfy the given conditions.

Learn how to find the value of x and the lengths AB, BC, and AC in a geometry problem where B is the midpoint of AC. Detailed step-by-step solution provided.

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