`triangle ABC` is an isosceles triangle with `AB = AC= 15 cm` and altitude from A to BC is 12 cm. The length of sides BC is :

Options

A. 9 cm
B. 12 cm
C. 18 cm
D. 20 cm

Answer

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

### Step 1: Understand the triangle configuration
Given that triangle ABC is an isosceles triangle with AB = AC = 15 cm and the altitude from A to BC is 12 cm, we can denote the foot of the altitude from A to BC as point D. This means that AD is perpendicular to BC.

### Step 2: Identify the segments
Since AD is the altitude, it divides BC into two equal segments. Let's denote BD = DC = x. Therefore, we can express BC as:
\[ BC = BD + DC = x + x = 2x \]

### Step 3: Apply the Pythagorean theorem
In triangle ABD, we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ 15^2 = 12^2 + x^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 15^2 = 225 \]
\[ 12^2 = 144 \]
So we have:
\[ 225 = 144 + x^2 \]

### Step 5: Solve for x^2
Rearranging the equation gives:
\[ x^2 = 225 - 144 \]
\[ x^2 = 81 \]

### Step 6: Find x
Taking the square root of both sides:
\[ x = \sqrt{81} = 9 \, \text{cm} \]

### Step 7: Calculate BC
Now, substituting back to find BC:
\[ BC = 2x = 2 \times 9 = 18 \, \text{cm} \]

### Final Answer
The length of side BC is **18 cm**.