In triangle `ABC, /_BAC = 90^@ `and AD is perpendicular to BC. If `AD = 6 cm `and `BD = 4 cm`, then the length of BC is:

Options

A. 10 cm
B. 12 cm
C. 13 cm
D. 15 cm.

Answer

To find the length of \( BC \) in triangle \( ABC \) where \( \angle BAC = 90^\circ \), \( AD \) is perpendicular to \( BC \), \( AD = 6 \, \text{cm} \), and \( BD = 4 \, \text{cm} \), we can follow these steps:

### Step 1: Understand the triangle configuration
We have triangle \( ABC \) with \( A \) at the top, \( B \) on the left, and \( C \) on the right. Since \( \angle BAC = 90^\circ \), \( AB \) is vertical and \( AC \) is horizontal. \( AD \) is the height from \( A \) to line \( BC \).

### Step 2: Identify the segments
Since \( D \) is the foot of the perpendicular from \( A \) to \( BC \), we know:
- \( AD = 6 \, \text{cm} \) (height from \( A \) to \( BC \))
- \( BD = 4 \, \text{cm} \) (part of \( BC \))

Let \( DC \) be the other part of \( BC \). Thus, we can express \( BC \) as:
\[ BC = BD + DC \]

### Step 3: Use the Pythagorean theorem
To find \( AC \) and \( AB \), we can apply the Pythagorean theorem in triangle \( ABD \):
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ AB^2 = 6^2 + 4^2 = 36 + 16 = 52 \]
Thus,
\[ AB = \sqrt{52} = 2\sqrt{13} \, \text{cm} \]

### Step 4: Calculate \( AC \) using the tangent function
Using the right triangle \( ABD \), we can find \( AC \) using the tangent function:
\[ \tan(\angle ABD) = \frac{AD}{BD} = \frac{6}{4} = \frac{3}{2} \]
This means that for every 2 units of \( BD \), \( AD \) rises 3 units. Therefore, we can express \( AC \) in terms of \( AB \):
\[ AC = \frac{3}{2} \times AB \]
Substituting \( AB = 2\sqrt{13} \):
\[ AC = \frac{3}{2} \times 2\sqrt{13} = 3\sqrt{13} \, \text{cm} \]

### Step 5: Find \( DC \)
Now, we can find \( DC \) using the Pythagorean theorem in triangle \( ACD \):
\[ AC^2 = AD^2 + DC^2 \]
Substituting the values:
\[ (3\sqrt{13})^2 = 6^2 + DC^2 \]
This simplifies to:
\[ 117 = 36 + DC^2 \]
Thus,
\[ DC^2 = 117 - 36 = 81 \implies DC = \sqrt{81} = 9 \, \text{cm} \]

### Step 6: Calculate \( BC \)
Now we can find the total length of \( BC \):
\[ BC = BD + DC = 4 + 9 = 13 \, \text{cm} \]

### Final Answer
The length of \( BC \) is \( 13 \, \text{cm} \).

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