In a triangle ABC with altitude `AD`, `angleBAC=45^(@), DB=3 and CD=2`. The area of the triangle `ABC` is : (a) 6 (b) 15 (c) `15/4` (d) 12

Options

A. 6
B. 15
C. `15//4`
D. 12

Answer

To find the area of triangle ABC with the given conditions, we can follow these steps:

### Step 1: Understand the Triangle
We have triangle ABC with altitude AD. Given:
- Angle BAC = 45°
- DB = 3
- CD = 2

### Step 2: Set Up the Diagram
Draw triangle ABC with point D on side BC such that:
- BD = 3
- CD = 2
- Therefore, BC = BD + CD = 3 + 2 = 5.

### Step 3: Define the Altitude
Let the length of altitude AD be denoted as \( h \).

### Step 4: Use the Tangent Function
Since angle BAC = 45°, we can express the tangent of the angles:
- Let angle BAD = \( \alpha \)
- Let angle CAD = \( \beta \)
- We know that \( \alpha + \beta = 45° \).

Using the tangent addition formula:
\[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} = 1 \]

### Step 5: Express Tangents in Terms of h
From triangle ABD:
\[ \tan \alpha = \frac{DB}{AD} = \frac{3}{h} \]
From triangle ACD:
\[ \tan \beta = \frac{CD}{AD} = \frac{2}{h} \]

### Step 6: Substitute into the Tangent Formula
Substituting the values into the tangent addition formula:
\[ \frac{\frac{3}{h} + \frac{2}{h}}{1 - \frac{3}{h} \cdot \frac{2}{h}} = 1 \]
This simplifies to:
\[ \frac{\frac{5}{h}}{1 - \frac{6}{h^2}} = 1 \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 5 = h - \frac{6}{h} \]
Multiplying through by \( h \) to eliminate the fraction:
\[ 5h = h^2 - 6 \]

### Step 8: Rearranging the Equation
Rearranging gives us a quadratic equation:
\[ h^2 - 5h - 6 = 0 \]

### Step 9: Factor the Quadratic
Factoring the quadratic:
\[ (h - 6)(h + 1) = 0 \]
This gives us two solutions:
\[ h = 6 \quad \text{or} \quad h = -1 \]
Since \( h \) represents a length, we take \( h = 6 \).

### Step 10: Calculate the Area
The area \( A \) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \( BC = 5 \) and height \( AD = h = 6 \):
\[ A = \frac{1}{2} \times 5 \times 6 = \frac{30}{2} = 15 \]

### Final Answer
The area of triangle ABC is \( 15 \).