In triangle ABC,` angle ABC - 90^(@) , AB = 2a+ 1 and BC = 2a^(2) +2a. ` Find AC in terms of 'a' if a= 8 , find the lengths of the sides of the triangles.

Answer

To find the length of side AC in triangle ABC, where angle ABC is 90 degrees, we can use the Pythagorean theorem. The lengths of sides AB and BC are given as follows:

- AB = 2a + 1
- BC = 2a² + 2a

### Step-by-Step Solution:

1. **Identify the sides**:
- We have AB = 2a + 1
- We have BC = 2a² + 2a

2. **Apply the Pythagorean theorem**:
According to the Pythagorean theorem, in a right triangle:
\[ AC^2 = AB^2 + BC^2 \]

3. **Calculate AB²**:
\[ AB^2 = (2a + 1)^2 = (2a)^2 + 2 \cdot (2a) \cdot 1 + 1^2 = 4a^2 + 4a + 1 \]

4. **Calculate BC²**:
\[ BC^2 = (2a^2 + 2a)^2 = (2a^2)^2 + 2 \cdot (2a^2) \cdot (2a) + (2a)^2 = 4a^4 + 8a^3 + 4a^2 \]

5. **Combine the results**:
Now substitute AB² and BC² into the Pythagorean theorem:
\[ AC^2 = (4a^2 + 4a + 1) + (4a^4 + 8a^3 + 4a^2) \]
Combine like terms:
\[ AC^2 = 4a^4 + 8a^3 + (4a^2 + 4a^2) + 4a + 1 = 4a^4 + 8a^3 + 8a^2 + 4a + 1 \]

6. **Find AC**:
To find AC, we take the square root:
\[ AC = \sqrt{4a^4 + 8a^3 + 8a^2 + 4a + 1} \]

7. **Substitute a = 8**:
Now substitute \( a = 8 \):
\[ AC = \sqrt{4(8^4) + 8(8^3) + 8(8^2) + 4(8) + 1} \]

Calculate each term:
- \( 8^4 = 4096 \)
- \( 8^3 = 512 \)
- \( 8^2 = 64 \)

Now substitute:
\[ AC = \sqrt{4(4096) + 8(512) + 8(64) + 4(8) + 1} \]
\[ = \sqrt{16384 + 4096 + 512 + 32 + 1} \]
\[ = \sqrt{21025} \]

8. **Calculate the square root**:
The square root of 21025 is:
\[ AC = 145 \]

9. **Calculate AB and BC**:
Now calculate AB and BC using \( a = 8 \):
- \( AB = 2(8) + 1 = 16 + 1 = 17 \)
- \( BC = 2(8^2) + 2(8) = 2(64) + 16 = 128 + 16 = 144 \)

### Final Results:
- AC = 145
- AB = 17
- BC = 144