In a right angled triangle ABC, `AB=2.5 "cm" cos B=0.5 /_ ACB= 90^(@)` Then length of side AC is?

Options

A. `5sqrt(3)`
B. `(5)/(2)sqrt(3)`
C. `(5)/(4)sqrt(3)`
D. `(5)/(16)sqrt(3)`

Answer

To find the length of side AC in triangle ABC, we can follow these steps:

### Step 1: Identify the Given Information
- We have a right triangle ABC with:
- \( AB = 2.5 \, \text{cm} \)
- \( \cos B = 0.5 \)
- \( \angle ACB = 90^\circ \)

### Step 2: Determine Angle B
- Since \( \cos B = 0.5 \), we can determine that:
- \( B = 60^\circ \) (because \( \cos 60^\circ = 0.5 \))

### Step 3: Calculate Angle A
- The sum of angles in a triangle is \( 180^\circ \). Therefore:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the known values:
\[ \angle A + 60^\circ + 90^\circ = 180^\circ \]
Simplifying gives:
\[ \angle A = 180^\circ - 150^\circ = 30^\circ \]

### Step 4: Use the Sine Rule
- The sine rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
In triangle ABC, we can write:
\[ \frac{AB}{\sin C} = \frac{AC}{\sin B} \]
Where:
- \( AB = 2.5 \, \text{cm} \)
- \( \sin C = \sin 90^\circ = 1 \)
- \( \sin B = \sin 60^\circ = \frac{\sqrt{3}}{2} \)

### Step 5: Substitute the Values into the Sine Rule
- Plugging in the values:
\[ \frac{2.5}{1} = \frac{AC}{\frac{\sqrt{3}}{2}} \]

### Step 6: Solve for AC
- Rearranging gives:
\[ AC = 2.5 \cdot \frac{\sqrt{3}}{2} \]
Simplifying:
\[ AC = \frac{2.5 \sqrt{3}}{2} = 1.25 \sqrt{3} \, \text{cm} \]

### Step 7: Final Calculation
- If we want to express AC in decimal form, using \( \sqrt{3} \approx 1.73 \):
\[ AC \approx 1.25 \cdot 1.73 \approx 2.165 \, \text{cm} \]

### Conclusion
- The length of side AC is \( 1.25 \sqrt{3} \, \text{cm} \) or approximately \( 2.165 \, \text{cm} \).

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