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.