In triangle ABC , `angleABC = 90^@ , angle CAB = x^@ , tan x^@ = 3/4 and BC = 15 cm `. Find the measures of AB and AC.

Answer

To solve the problem step by step, we will follow the information given and apply trigonometric ratios and the Pythagorean theorem.

### Step 1: Understand the triangle and given values
We have triangle ABC where:
- Angle ABC = 90°
- Angle CAB = x°
- tan(x) = 3/4
- Length of BC = 15 cm

### Step 2: Use the tangent ratio
From the definition of tangent in a right triangle:
\[ \tan(x) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Given that \( \tan(x) = \frac{3}{4} \), we can set up the equation:
\[ \frac{AB}{BC} = \frac{3}{4} \]
Substituting \( BC = 15 \) cm into the equation:
\[ \frac{AB}{15} = \frac{3}{4} \]

### Step 3: Solve for AB
Cross-multiplying gives:
\[ 4 \cdot AB = 3 \cdot 15 \]
\[ 4 \cdot AB = 45 \]
Now, divide both sides by 4:
\[ AB = \frac{45}{4} = 11.25 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find AC
In triangle ABC, we can use the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values we have:
\[ AC^2 = (11.25)^2 + (15)^2 \]
Calculating \( (11.25)^2 \):
\[ (11.25)^2 = 126.5625 \]
Calculating \( (15)^2 \):
\[ (15)^2 = 225 \]
Now, adding these values:
\[ AC^2 = 126.5625 + 225 = 351.5625 \]

### Step 5: Find AC
Taking the square root of both sides:
\[ AC = \sqrt{351.5625} \approx 18.75 \text{ cm} \]

### Final Answers
- Length of AB = 11.25 cm
- Length of AC = 18.75 cm