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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given information
We have triangle ABC with:
- Angle B = 90°
- AB = 40 (which is the side opposite to angle C)
- AC + BC = 80 (the sum of the other two sides)

### Step 2: Assign variables
Let:
- AC = x
- BC = 80 - x (since AC + BC = 80)

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem, in a right triangle:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ 40^2 + (80 - x)^2 = x^2 \]

### Step 4: Expand the equation
Calculating \( 40^2 \):
\[ 1600 + (80 - x)^2 = x^2 \]
Now expand \( (80 - x)^2 \):
\[ 1600 + (6400 - 160x + x^2) = x^2 \]

### Step 5: Simplify the equation
Combine like terms:
\[ 1600 + 6400 - 160x + x^2 = x^2 \]
The \( x^2 \) terms cancel out:
\[ 8000 - 160x = 0 \]

### Step 6: Solve for x
Rearranging gives:
\[ 160x = 8000 \]
Dividing both sides by 160:
\[ x = \frac{8000}{160} = 50 \]

### Step 7: Find the lengths of AC and BC
Now we can find:
- AC = x = 50
- BC = 80 - x = 80 - 50 = 30

### Step 8: Calculate cos A
Using the definition of cosine in a right triangle:
\[ \cos A = \frac{BC}{AC} \]
Substituting the values we found:
\[ \cos A = \frac{30}{50} = \frac{3}{5} \]

### Final Answer
Thus, the value of \( \cos A \) is \( \frac{3}{5} \).

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