Let's solve the problem step by step, based on the image you provided.
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Problem Statement:
Given:
- $ AB = 25 $
- Point $ N $ lies on segment $ AB $
- $ AN = 2x - 6 $
- $ NB = x + 7 $
We are asked to:
1. Find the value of $ x $
2. Find the lengths of $ AN $ and $ NB $
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Step 1: Use the Segment Addition Postulate
The
Segment Addition Postulate states that if point $ N $ is between points $ A $ and $ B $, then:
$$
AN + NB = AB
$$
Substitute the given expressions:
$$
(2x - 6) + (x + 7) = 25
$$
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Step 2: Simplify the Equation
Combine like terms on the left side:
$$
2x + x - 6 + 7 = 25 \\
3x + 1 = 25
$$
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Step 3: Solve for $ x $
Subtract 1 from both sides:
$$
3x = 24
$$
Divide both sides by 3:
$$
x = 8
$$
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Step 4: Find $ AN $ and $ NB $
Now substitute $ x = 8 $ into the expressions:
- $ AN = 2x - 6 = 2(8) - 6 = 16 - 6 = 10 $
- $ NB = x + 7 = 8 + 7 = 15 $
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Step 5: Check the Answer
Add $ AN $ and $ NB $:
$$
AN + NB = 10 + 15 = 25
$$
This matches $ AB = 25 $, so the solution checks out.
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✔ Final Answers:
- $ x = 8 $
- $ AN = 10 $
- $ NB = 15 $
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🔍 Explanation Summary:
We used the
Segment Addition Postulate, which allows us to write an equation based on the total length of a segment being equal to the sum of its parts. By substituting the algebraic expressions for the segments, simplifying, and solving for $ x $, we found the unknown value. Then we substituted back to find the actual lengths of $ AN $ and $ NB $. The final check confirms our answer is correct.
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✔ Answer:
- $ x = 8 $
- $ AN = 10 $
- $ NB = 15 $
Parent Tip: Review the logic above to help your child master the concept of measuring segments.
