In triangle ABC ,` angle B = 90 ^(@) , AB = 40 , AC + BC = 80 , ` Find :
tan C .

Answer

To solve the problem, we will follow these steps:

1. **Draw Triangle ABC**: We have a right triangle ABC where angle B = 90°. We know that AB = 40 and AC + BC = 80.

2. **Assign Variables**: Let AC = x. Then, BC can be expressed as BC = 80 - x.

3. **Apply the Pythagorean Theorem**: According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC). Therefore, we can write:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ 40^2 + (80 - x)^2 = x^2 \]

4. **Simplify the Equation**: Now, we will simplify the equation:
\[ 1600 + (80 - x)^2 = x^2 \]
Expanding \((80 - x)^2\) using the formula \(a^2 - 2ab + b^2\):
\[ 1600 + (6400 - 160x + x^2) = x^2 \]
This simplifies to:
\[ 8000 - 160x + x^2 = x^2 \]

5. **Cancel x² and Rearrange**: The \(x^2\) terms on both sides cancel out:
\[ 8000 - 160x = 0 \]
Rearranging gives:
\[ 160x = 8000 \]

6. **Solve for x**: Dividing both sides by 160:
\[ x = \frac{8000}{160} = 50 \]

7. **Find BC**: Now that we have AC = x = 50, we can find BC:
\[ BC = 80 - AC = 80 - 50 = 30 \]

8. **Calculate tan C**: We need to find \(\tan C\):
\[ \tan C = \frac{AB}{BC} = \frac{40}{30} = \frac{4}{3} \]

Thus, the final answer is:
\[ \tan C = \frac{4}{3} \]