In triangle ABC, `angleB = 90^@` and tan A = 0.75 If AC = 30 cm, find the lengths of AB and BC.

Answer

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

### Step 1: Understand the triangle and given information
We have a right triangle ABC where angle B = 90°. We are given:
- tan A = 0.75
- AC = 30 cm

### Step 2: Set up the relationship using the tangent function
In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For angle A:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} \]
Given that \( \tan A = 0.75 \), we can express this as:
\[ \frac{BC}{AB} = 0.75 \]
This can be rewritten as:
\[ BC = 0.75 \times AB \]

### Step 3: Express BC in terms of AB
To make calculations easier, we can express 0.75 as a fraction:
\[ 0.75 = \frac{3}{4} \]
Thus:
\[ BC = \frac{3}{4} AB \]

### Step 4: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting \( AC = 30 \) cm:
\[ 30^2 = AB^2 + BC^2 \]
This simplifies to:
\[ 900 = AB^2 + BC^2 \]

### Step 5: Substitute BC in the Pythagorean theorem
Now substitute \( BC = \frac{3}{4} AB \) into the equation:
\[ 900 = AB^2 + \left(\frac{3}{4} AB\right)^2 \]
Calculating \( \left(\frac{3}{4} AB\right)^2 \):
\[ \left(\frac{3}{4} AB\right)^2 = \frac{9}{16} AB^2 \]
So the equation becomes:
\[ 900 = AB^2 + \frac{9}{16} AB^2 \]

### Step 6: Combine like terms
Combine the terms on the right side:
\[ 900 = \left(1 + \frac{9}{16}\right) AB^2 \]
Convert 1 to a fraction:
\[ 1 = \frac{16}{16} \]
Thus:
\[ 900 = \left(\frac{16}{16} + \frac{9}{16}\right) AB^2 \]
\[ 900 = \frac{25}{16} AB^2 \]

### Step 7: Solve for AB
To isolate \( AB^2 \), multiply both sides by \( \frac{16}{25} \):
\[ AB^2 = 900 \times \frac{16}{25} \]
Calculating the right side:
\[ AB^2 = \frac{14400}{25} = 576 \]
Taking the square root:
\[ AB = \sqrt{576} = 24 \text{ cm} \]

### Step 8: Find BC using AB
Now that we have \( AB \), we can find \( BC \):
\[ BC = \frac{3}{4} AB = \frac{3}{4} \times 24 = 18 \text{ cm} \]

### Final Answer
- Length of AB = 24 cm
- Length of BC = 18 cm