In `DeltaABC`, right angled at B, AB = 24 cm, BC = 7 cm. The value of `sin C` is :

Options

A. `(24)/(25)`
B. `(7)/(25)`
C. `(7)/(24)`
D. None of these

Answer

To find the value of \( \sin C \) in triangle \( \Delta ABC \) where \( B \) is the right angle, \( AB = 24 \) cm, and \( BC = 7 \) cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
In triangle \( ABC \):
- \( AB \) is the side opposite angle \( C \) (perpendicular).
- \( BC \) is the side adjacent to angle \( C \) (base).
- \( AC \) is the hypotenuse.

### Step 2: Use the Pythagorean theorem to find the hypotenuse \( AC \)
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values:
\[ AC^2 = 24^2 + 7^2 \]
Calculating the squares:
\[ AC^2 = 576 + 49 \]
\[ AC^2 = 625 \]
Now, take the square root to find \( AC \):
\[ AC = \sqrt{625} = 25 \text{ cm} \]

### Step 3: Calculate \( \sin C \)
The sine of angle \( C \) is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
\[ \sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} \]
Substituting the known values:
\[ \sin C = \frac{24}{25} \]

### Final Answer
Thus, the value of \( \sin C \) is:
\[ \sin C = \frac{24}{25} \]

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