The problem provided is to find the length of $EG$ on a line segment involving several sub-segments. Here's what we have from the image:

• $E$ to $F$ has a length of $x + 23$,
• $F$ to $G$ has a length of $14 + x$,
• $G$ to $H$ has a length of 12,
• The total length from $E$ to $H$ is 26.

We want to find $EG$, which is the length from $E$ to $G$.

Step 1: Express the total length of $EH$

The total length $EH$ is: $EH = EF + FG + GH$ Substituting the given values: $26 = (x + 23) + (14 + x) + 12$

Step 2: Simplify and solve for $x$

Simplify the equation: $26 = (x + 23) + (14 + x) + 12$ $26 = 2x + 23 + 14 + 12$ $26 = 2x + 49$ Now solve for $x$: $26 - 49 = 2x$ $-23 = 2x$ $x = -\frac{23}{2} = -11.5$

Step 3: Find $EG$

Now, we can find $EG$. The length $EG$ is the sum of $EF$ and $FG$, so: $EG = (x + 23) + (14 + x)$ Substitute $x = -11.5$: $EG = (-11.5 + 23) + (14 - 11.5)$ $EG = 11.5 + 2.5 = 14$

Thus, the length of $EG$ is $14$.

Let me know if you need further details or have any questions!

Related Questions:

1. How would the problem change if the total length of $EH$ was 30 instead of 26?
2. What would be the value of $x$ if the segment $GH$ had a different length?
3. How can we use algebraic expressions to represent line segments in geometry problems?
4. Why is it necessary to solve for $x$ before calculating the length of $EG$?
5. How can this type of problem help in understanding coordinate geometry?

Tip:

Always organize your information clearly when solving geometry problems involving multiple line segments!