To determine the length of [tex]\(\overline{GQ}\)[/tex], let's follow these steps:
1. Identify the given expressions:
- [tex]\(GQ = 2x + 3\)[/tex]
- [tex]\(GH = 5x - 5\)[/tex]
2. Understand the relationship between [tex]\(GQ\)[/tex] and [tex]\(GH\)[/tex]:
Since point [tex]\(Q\)[/tex] is the midpoint of [tex]\(\overline{GH}\)[/tex], the length [tex]\(GQ\)[/tex] is half the length of [tex]\(\overline{GH}\)[/tex]. Mathematically, this relationship is given by:
[tex]\[
GQ = \frac{1}{2} \times GH
\][/tex]
3. Substitute the given expressions:
[tex]\[
2x + 3 = \frac{1}{2} \times (5x - 5)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
First, eliminate the fraction by multiplying both sides of the equation by 2:
[tex]\[
2 \times (2x + 3) = 5x - 5
\][/tex]
Simplifying, we get:
[tex]\[
4x + 6 = 5x - 5
\][/tex]
Next, isolate [tex]\(x\)[/tex]:
[tex]\[
6 + 5 = 5x - 4x
\][/tex]
[tex]\[
11 = x
\][/tex]
5. Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(GQ\)[/tex] to find its length:
[tex]\[
GQ = 2x + 3
\][/tex]
Substitute [tex]\(x = 11\)[/tex]:
[tex]\[
GQ = 2(11) + 3
\][/tex]
[tex]\[
GQ = 22 + 3
\][/tex]
[tex]\[
GQ = 25
\][/tex]
Therefore, the length of [tex]\(\overline{GQ}\)[/tex] is [tex]\(25\)[/tex] units.
