In a triangle ABC, the two sides AB and AC are 36 cm and 30 cm, and the radius of its circumcircle is 45 cm. what's the length of perpendicular drawn from its vertex A on the sides BC?

Options

A. A)12 cm
B. B)18 cm
C. C)15 cm
D. D)33 cm

Answer

To find the length of the perpendicular drawn from vertex A to side BC in triangle ABC, we can use the relationship between the circumradius (R), the sides of the triangle, and the area of the triangle.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Length of side AB (c) = 36 cm
- Length of side AC (b) = 30 cm
- Radius of circumcircle (R) = 45 cm
- Let the length of side BC (a) = K (unknown)

2. **Use the Formula for Circumradius**:
The formula for the circumradius (R) of a triangle is given by:
\[ R = \frac{abc}{4 \times \text{Area}} \]
where \( a, b, c \) are the lengths of the sides of the triangle.

3. **Express Area in Terms of Base and Height**:
The area of triangle ABC can also be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take BC (K) as the base and the height from A to BC as AD (which we need to find). Thus:
\[ \text{Area} = \frac{1}{2} \times K \times AD \]

4. **Substitute Area into the Circumradius Formula**:
Substitute the expression for the area into the circumradius formula:
\[ R = \frac{abc}{2K \times AD} \]
Rearranging gives:
\[ AD = \frac{abc}{2RK} \]

5. **Substitute Known Values**:
Substitute \( R = 45 \), \( b = 30 \), \( c = 36 \):
\[ AD = \frac{36 \times 30 \times K}{2 \times 45 \times K} \]
Notice that K cancels out:
\[ AD = \frac{36 \times 30}{2 \times 45} \]

6. **Calculate the Length of AD**:
Simplifying the expression:
\[ AD = \frac{1080}{90} = 12 \text{ cm} \]

### Final Answer:
The length of the perpendicular drawn from vertex A to side BC (AD) is **12 cm**.