In `DeltaABC`, angle B is right angle, AB = 12 cm and BC = 5 m. The value of sin C is :

Options

A. `12/13`
B. `5/13`
C. `12/5`
D. `5/12`

Answer

To find the value of sin C in triangle ABC, where angle B is a right angle, we can follow these steps:

### Step 1: Identify the sides of the triangle
In triangle ABC:
- AB = 12 cm (one leg of the triangle)
- BC = 5 cm (the other leg of the triangle)
- Angle B = 90 degrees (right angle)

### Step 2: Use the Pythagorean theorem to find the hypotenuse AC
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]

Substituting the known values:
\[ AC^2 = 12^2 + 5^2 \]
\[ AC^2 = 144 + 25 \]
\[ AC^2 = 169 \]

Now, take the square root to find AC:
\[ AC = \sqrt{169} = 13 \, \text{cm} \]

### Step 3: Use the definition of sine to find sin C
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

For angle C:
- Opposite side = AB = 12 cm
- Hypotenuse = AC = 13 cm

Thus,
\[ \sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{12}{13} \]

### Final Answer
The value of sin C is:
\[ \sin C = \frac{12}{13} \]

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