In a triangle ABC, BC = 5 cm, AC = 12 cm and AB = 13 cm. The length of the altitude drawn from B on AC is -----

Options

A. 4 cm
B. 5 cm
C. 6 cm
D. 7 cm

Answer

To find the length of the altitude drawn from point B to side AC in triangle ABC, we can follow these steps:

### Step 1: Identify the triangle properties
We have a triangle ABC with sides:
- BC = 5 cm
- AC = 12 cm
- AB = 13 cm

### Step 2: Check if the triangle is a right triangle
We can check if triangle ABC is a right triangle using the Pythagorean theorem. According to the theorem, for a triangle with sides a, b, and c (where c is the hypotenuse), the relationship should hold:
\[ c^2 = a^2 + b^2 \]

In our case:
- Let AB = c = 13 cm (hypotenuse)
- BC = a = 5 cm
- AC = b = 12 cm

Calculating:
\[ 13^2 = 5^2 + 12^2 \]
\[ 169 = 25 + 144 \]
\[ 169 = 169 \]

Since the equation holds true, triangle ABC is a right triangle with the right angle at B.

### Step 3: Calculate the area of triangle ABC
The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

In this triangle, we can take AC as the base and BC as the height:
\[ A = \frac{1}{2} \times AC \times BC \]
\[ A = \frac{1}{2} \times 12 \times 5 \]
\[ A = \frac{1}{2} \times 60 \]
\[ A = 30 \text{ cm}^2 \]

### Step 4: Use the area to find the altitude from B to AC
The area can also be expressed using the base AC and the height (altitude from B to AC):
\[ A = \frac{1}{2} \times AC \times h \]
Where \( h \) is the altitude from B to AC.

Setting the two area expressions equal:
\[ 30 = \frac{1}{2} \times 12 \times h \]

### Step 5: Solve for h
To find \( h \), rearranging the equation gives:
\[ 30 = 6h \]
\[ h = \frac{30}{6} \]
\[ h = 5 \text{ cm} \]

Thus, the length of the altitude drawn from B to AC is **5 cm**.

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