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The problem states that two segments, \( AB \) and \( CD \), have lengths of 8 and \( a \), respectively. Additionally, two other segments, \( EF \) and \( GH \), share the same ratio as \( AB \) and \( CD \), with lengths of 10 and \( a + 10 \), respectively. We are tasked with finding the value of \( a \).

### Step-by-Step Solution:

1. The ratio between \( AB \) and \( CD \) is:
   \[
   \frac{AB}{CD} = \frac{8}{a}
   \]

2. Similarly, the ratio between \( EF \) and \( GH \) is:
   \[
   \frac{EF}{GH} = \frac{10}{a + 10}
   \]

3. Since the two ratios are equal, we can set them equal to each other:
   \[
   \frac{8}{a} = \frac{10}{a + 10}
   \]

4. To solve for \( a \), cross-multiply:
   \[
   8(a + 10) = 10a
   \]

5. Distribute on the left side:
   \[
   8a + 80 = 10a
   \]

6. Now, solve for \( a \) by subtracting \( 8a \) from both sides:
   \[
   80 = 2a
   \]

7. Divide both sides by 2:
   \[
   a = 40
   \]

Thus, the value of \( a \) is \( \boxed{40} \).

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Would you like further details on the steps, or do you have any questions?

Here are 5 related questions to expand on this topic:
1. What would happen if the ratio between the segments was reversed?
2. How can we verify that \( a = 40 \) satisfies the ratio conditions?
3. What is the general form for setting up proportions in similar geometric problems?
4. Can we generalize this problem to other similar figures with different values?
5. How does cross-multiplication help in solving proportion equations?

**Tip:** Always double-check your cross-multiplication when dealing with ratios to avoid simple errors!

AB and CD are two segments. Their lengths are 8 and a, respectively. EF and GH share the same ratio as AB and CD; however, EF has a length of 10 and GH has a length of (a + 10). Find the value of a.

Learn how to solve a segment ratio problem where AB and CD have lengths of 8 and a, and EF and GH share the same ratio. This problem involves algebraic cross-multiplication and proportional reasoning, suitable for students in Grades 8-10.

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