In a triangle ABC, right-angled at B, side BC = 20 cm and angle A= 30°. Find the length of AB.

Answer

To solve the problem step-by-step, we will use the properties of right triangles and trigonometric ratios.

### Step 1: Understand the Triangle
We have a right triangle ABC, where:
- Angle B is the right angle (90°).
- Angle A is given as 30°.
- Side BC (the side opposite to angle A) is given as 20 cm.

### Step 2: Find Angle C
Using the property that the sum of angles in a triangle is 180°, we can find angle C:
\[ \text{Angle C} = 180° - \text{Angle A} - \text{Angle B} = 180° - 30° - 90° = 60° \]

### Step 3: Identify the Sides
In triangle ABC:
- AB is the side we want to find (let's denote it as \( x \)).
- BC is the side opposite angle A, which is 20 cm.
- AC is the hypotenuse of the triangle.

### Step 4: Use the Tangent Function
Since we know angle A (30°) and the opposite side (BC), we can use the tangent function:
\[ \tan(A) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} \]
This can be rewritten as:
\[ \tan(30°) = \frac{20}{x} \]

### Step 5: Substitute the Value of Tangent
We know that:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \quad \text{(or approximately 0.577)} \]
So we can substitute this into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{20}{x} \]

### Step 6: Cross Multiply to Solve for x
Cross multiplying gives us:
\[ x = 20 \sqrt{3} \]

### Step 7: Calculate the Approximate Value
To find the numerical value, we can use the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ x \approx 20 \times 1.732 = 34.64 \text{ cm} \]

### Final Answer
The length of AB is \( 20\sqrt{3} \) cm or approximately 34.64 cm.

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