A circle of radius 4 cm is inscribed in `DeltaABC`, which touches side BC at D. If BD = 6 cm, DC = 8 cm then

Options

A. the triangle is necessarily acute angled
B. `tan.(A)/(2) = (4)/(7)`
C. perimeter of the triangle ABC is 42 cm
D. area of `DeltaABC " is " 84 cm^(2)`

Explanation

To solve the problem step by step, we will follow the logical progression outlined in the video transcript.

### Step-by-Step Solution:

1. **Draw the Triangle and Identify Points**:
- Draw triangle ABC with an inscribed circle that touches side BC at point D.
- Label the lengths: BD = 6 cm and DC = 8 cm.

2. **Calculate Length of Side BC**:
- Since BD + DC = BC, we can calculate:
\[ BC = BD + DC = 6 \text{ cm} + 8 \text{ cm} = 14 \text{ cm} \]

3. **Identify the Inradius (r)**:
- The radius of the inscribed circle (inradius) is given as r = 4 cm.

4. **Use the Incenter Angles**:
- Let I be the incenter of triangle ABC. The angles at I are related to the angles of triangle ABC:
- \(\angle IBD = \frac{B}{2}\)
- \(\angle ICD = \frac{C}{2}\)

5. **Calculate Tangents**:
- Using the tangent definitions:
\[ \tan \left(\frac{B}{2}\right) = \frac{r}{BD} = \frac{4}{6} = \frac{2}{3} \]
\[ \tan \left(\frac{C}{2}\right) = \frac{r}{DC} = \frac{4}{8} = \frac{1}{2} \]

6. **Use the Formula for Tangents**:
- We use the formula for the product of tangents:
\[ \tan \left(\frac{B}{2}\right) \tan \left(\frac{C}{2}\right) = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}} \]
- Here, \(s\) is the semi-perimeter of the triangle.

7. **Calculate the Semi-perimeter (s)**:
- The semi-perimeter \(s\) is given by:
\[ s = \frac{AB + AC + BC}{2} \]
- We need to express \(s\) in terms of \(a\), \(b\), and \(c\). We know \(BC = a = 14\) cm.

8. **Set Up the Equation**:
- Let \(b = AB\) and \(c = AC\). We can express the tangents:
\[ \frac{2}{3} \cdot \frac{1}{2} = \sqrt{\frac{(s-a)(s-b)}{s(s-c)}} \]
- Simplifying gives:
\[ \frac{1}{3} = \sqrt{\frac{(s-14)(s-b)}{s(s-c)}} \]

9. **Substituting Values**:
- We know \(s = \frac{AB + AC + 14}{2}\). We can solve for \(s\) using the area formula.

10. **Calculate Area of Triangle**:
- The area \(A\) of triangle ABC can be calculated using:
\[ A = r \cdot s \]
- We already know \(r = 4\) cm. We need to find \(s\) first.

11. **Final Calculation**:
- From the previous steps, we can find \(s\) to be 21 cm.
- Therefore, the area is:
\[ A = 4 \cdot 21 = 84 \text{ cm}^2 \]

### Final Answer:
The area of triangle ABC is \(84 \text{ cm}^2\).