In a triangle ABC. AC = 5 cm, AB = 7 cm, `BC =sqrt39` what is the measure of angle A ? is the measure of angle A ?

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `30^(@)`

Answer

To find the measure of angle A in triangle ABC where AC = 5 cm, AB = 7 cm, and BC = √39 cm, we can use the cosine rule. The cosine rule states:

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

Where:
- \( a \) is the side opposite angle A (BC),
- \( b \) is the side opposite angle B (AC),
- \( c \) is the side opposite angle C (AB).

In our case:
- \( a = \sqrt{39} \) (BC),
- \( b = 5 \) (AC),
- \( c = 7 \) (AB).

### Step 1: Substitute the values into the cosine rule formula

\[ \cos A = \frac{5^2 + 7^2 - (\sqrt{39})^2}{2 \cdot 5 \cdot 7} \]

### Step 2: Calculate \( 5^2 \), \( 7^2 \), and \( (\sqrt{39})^2 \)

\[ 5^2 = 25 \]
\[ 7^2 = 49 \]
\[ (\sqrt{39})^2 = 39 \]

### Step 3: Substitute these values back into the formula

\[ \cos A = \frac{25 + 49 - 39}{2 \cdot 5 \cdot 7} \]

### Step 4: Simplify the numerator

\[ 25 + 49 = 74 \]
\[ 74 - 39 = 35 \]

### Step 5: Calculate the denominator

\[ 2 \cdot 5 \cdot 7 = 70 \]

### Step 6: Substitute the simplified numerator and denominator

\[ \cos A = \frac{35}{70} \]

### Step 7: Simplify the fraction

\[ \cos A = \frac{1}{2} \]

### Step 8: Find the angle A

The angle whose cosine is \( \frac{1}{2} \) is \( 60^\circ \).

### Conclusion

Thus, the measure of angle A is \( 60^\circ \).

---