If a, b and c are three natural numbers in ascending order, then

Options

A. `c^(2) -a^(2) =b^(2)`
B. `c^(2) -a^(2) lt b^(2)`
C. `c^(2) + b^(2) =a^(2)`
D. `c^(2) -a^(2) gt b`

Answer

To solve the problem, we need to analyze the given options based on the condition that \( a \), \( b \), and \( c \) are three natural numbers in ascending order, meaning \( a < b < c \).

Let's assume three natural numbers:
- Let \( a = 1 \)
- Let \( b = 2 \)
- Let \( c = 3 \)

Now, we will evaluate each option:

### Option A: \( c^2 - a^2 = b^2 \)
1. Calculate \( c^2 \): \( 3^2 = 9 \)
2. Calculate \( a^2 \): \( 1^2 = 1 \)
3. Calculate \( b^2 \): \( 2^2 = 4 \)
4. Substitute into the equation: \( 9 - 1 = 4 \)
5. This simplifies to \( 8 = 4 \), which is **false**.

### Option B: \( c^2 - a^2 < b^2 \)
1. We already calculated \( c^2 - a^2 = 9 - 1 = 8 \).
2. We know \( b^2 = 4 \).
3. Substitute into the inequality: \( 8 < 4 \), which is **false**.

### Option C: \( c^2 + b^2 = a^2 \)
1. We already calculated \( c^2 = 9 \) and \( b^2 = 4 \).
2. Substitute into the equation: \( 9 + 4 = 1 \).
3. This simplifies to \( 13 = 1 \), which is **false**.

### Option D: \( c^2 - a^2 > b^2 \)
1. We already calculated \( c^2 - a^2 = 8 \).
2. We know \( b^2 = 4 \).
3. Substitute into the inequality: \( 8 > 4 \), which is **true**.

### Conclusion
The correct option is **D**: \( c^2 - a^2 > b^2 \).

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