In `DeltaABC,mangleB=90^(@),AB=4sqrt(5).BDbotAc,AD=4` , then area of `(DeltaABC)` =

Options

A. 96 sq . Units
B. 80 sq . Units
C. 120 sq . Units
D. 160 sq . Units

Answer

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

### Step 1: Understand the Given Information
- We have triangle ABC where angle B = 90°.
- AB = 4√5
- AD = 4
- BD is perpendicular to AC.

### Step 2: Use the Pythagorean Theorem
Since triangle ABD is a right triangle, we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ (4\sqrt{5})^2 = 4^2 + BD^2 \]

### Step 3: Calculate the Squares
Calculating the squares:
- \( (4\sqrt{5})^2 = 16 \times 5 = 80 \)
- \( AD^2 = 4^2 = 16 \)

Now, substituting these values into the equation:
\[ 80 = 16 + BD^2 \]

### Step 4: Solve for BD
Rearranging the equation to find BD:
\[ BD^2 = 80 - 16 \]
\[ BD^2 = 64 \]
Taking the square root:
\[ BD = \sqrt{64} = 8 \]

### Step 5: Find DC Using Similar Triangles
Using the property of similar triangles (triangles ABD and BDC):
\[ \frac{AD}{BD} = \frac{BD}{DC} \]
Substituting the known values:
\[ \frac{4}{8} = \frac{8}{DC} \]

### Step 6: Cross-Multiply to Solve for DC
Cross-multiplying gives:
\[ 4 \cdot DC = 8 \cdot 8 \]
\[ 4 \cdot DC = 64 \]
Dividing both sides by 4:
\[ DC = \frac{64}{4} = 16 \]

### Step 7: Find AC
Now, we can find AC:
\[ AC = AD + DC = 4 + 16 = 20 \]

### Step 8: Calculate the Area of Triangle ABC
The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, base AC = 20 and height BD = 8:
\[ \text{Area} = \frac{1}{2} \times 20 \times 8 \]

### Step 9: Simplify the Area Calculation
Calculating the area:
\[ \text{Area} = \frac{1}{2} \times 20 \times 8 = 10 \times 8 = 80 \]

### Final Answer
The area of triangle ABC is \( 80 \) square units.

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