In `DeltaABC,angleA=90^(@),AB=16" cm and "AC=12" cm"=12" cm"`. D is the midpoint of AC and `DE bot CB` at E. What is the area `("in cm"^(2))` of `Delta` CDE ?

Options

A. 8.64
B. 7.68
C. 5.76
D. 6.25

Answer

To solve the problem, we need to find the area of triangle CDE given the triangle ABC with the provided dimensions.

### Step-by-Step Solution:

1. **Identify the Triangle ABC**:
- We have triangle ABC where angle A is 90 degrees, AB = 16 cm, and AC = 12 cm.

2. **Calculate the Length of BC**:
- Using the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]
\[ BC^2 = 16^2 + 12^2 = 256 + 144 = 400 \]
\[ BC = \sqrt{400} = 20 \text{ cm} \]

3. **Find the Midpoint D of AC**:
- Since D is the midpoint of AC, we can find the coordinates of D. If A is at (0, 0) and C is at (12, 0), then:
\[ D = \left(\frac{0 + 12}{2}, \frac{0 + 0}{2}\right) = (6, 0) \]

4. **Determine the Coordinates of B**:
- Since AB = 16 cm and AC = 12 cm, B will be at (0, 16) (as it is vertically above A).

5. **Find the Equation of Line BC**:
- The slope of line BC can be calculated as follows:
\[ \text{slope of BC} = \frac{y_B - y_C}{x_B - x_C} = \frac{16 - 0}{0 - 12} = -\frac{4}{3} \]
- The equation of line BC in point-slope form (using point B (0, 16)):
\[ y - 16 = -\frac{4}{3}(x - 0) \implies y = -\frac{4}{3}x + 16 \]

6. **Find the Coordinates of E**:
- DE is perpendicular to BC, so the slope of DE will be the negative reciprocal of the slope of BC:
\[ \text{slope of DE} = \frac{3}{4} \]
- The equation of line DE using point D (6, 0):
\[ y - 0 = \frac{3}{4}(x - 6) \implies y = \frac{3}{4}x - \frac{9}{2} \]
- To find E, set the equations of lines BC and DE equal to each other:
\[ -\frac{4}{3}x + 16 = \frac{3}{4}x - \frac{9}{2} \]
- Solving this equation gives us the x-coordinate of E.

7. **Calculate the Area of Triangle CDE**:
- The area of triangle CDE can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Here, CD will be the base, and DE will be the height.

8. **Final Calculation**:
- Substitute the appropriate values into the area formula to find the area of triangle CDE.