Point Q is the midpoint of GH¯¯¯¯¯¯. GQ=2x+3, and GH=5x−5 .

What is the length of GQ¯¯¯¯¯?

Since point \( Q \) is the midpoint of \( \overline{GH} \), the length of \( \overline{GQ} \) is half the length of \( \overline{GH} \). 

Given:
- \( GQ = 2x + 3 \)
- \( GH = 5x - 5 \)

We know that:

\[
GQ = \frac{GH}{2}
\]

Substitute the expression for \( GH \):

\[
2x + 3 = \frac{5x - 5}{2}
\]

Now solve for \( x \):

1. Multiply both sides by 2 to eliminate the fraction:
   \[
   2(2x + 3) = 5x - 5
   \]
   \[
   4x + 6 = 5x - 5
   \]

2. Subtract \( 4x \) from both sides:
   \[
   6 = x - 5
   \]

3. Add 5 to both sides:
   \[
   x = 11
   \]

Now substitute \( x = 11 \) into the expression for \( GQ \):

\[
GQ = 2(11) + 3 = 22 + 3 = 25
\]

Thus, the length of \( \overline{GQ} \) is 25 units.

Would you like more details or have any questions?

Here are some related questions for further practice:
1. If \( GH = 30 \), what is the value of \( GQ \)?
2. How do you find the midpoint between two points on a number line?
3. Can you find \( x \) if \( GH = 4x + 6 \) and \( GQ = 2x - 1 \)?
4. What are the properties of a midpoint in a line segment?
5. How would you solve if \( GQ = 3x + 2 \) and \( GH = 6x - 8 \)?

**Tip:** Always check the relationship between segment parts when given a midpoint!

Point Q is the midpoint of GH. GQ = 2x + 3, and GH = 5x − 5. What is the length of GQ?

Learn how to solve a problem involving the midpoint of a line segment, where GQ = 2x + 3 and GH = 5x - 5. This step-by-step solution involves algebraic manipulation of linear equations and is suitable for students in Grades 8-10.

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