The hypotenuse AC of a right angle triangle ABC is 42 cm. this triangle has the maximum possible area with the given hypotenuse. A rectangle EFGH is formed in such a way that the two vertices G and H lie on the hypotenuse AC while the vertices E and F lie on BC and AB, respectively. if EH=14 cm, find the area of the largest possible circle that can be inscribed in the rectangle EFGH.

Options

A. 155 sq cm
B. 140 sq cm
C. 154 sq cm
D. data insufficient

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Geometry
We have a right-angled triangle ABC with hypotenuse AC measuring 42 cm. A rectangle EFGH is inscribed in this triangle such that vertices G and H lie on the hypotenuse AC, while vertices E and F lie on sides BC and AB, respectively.

### Step 2: Identify Given Information
- Hypotenuse AC = 42 cm
- EH = 14 cm (which is the height of the rectangle from the base AC)

### Step 3: Determine the Dimensions of the Rectangle
Since EH = 14 cm, the opposite side FG will also be 14 cm (because opposite sides of a rectangle are equal). Therefore, the breadth of rectangle EFGH is 14 cm.

### Step 4: Find the Radius of the Inscribed Circle
The radius of the circle inscribed in rectangle EFGH is half of the breadth (since the circle will touch the sides of the rectangle).
- Radius (r) = EH / 2 = 14 cm / 2 = 7 cm

### Step 5: Calculate the Area of the Circle
The area of a circle is given by the formula:
\[ \text{Area} = \pi r^2 \]
Substituting the radius we found:
\[ \text{Area} = \pi (7)^2 = \pi \times 49 \]

Using the approximate value of \(\pi \approx 3.14\):
\[ \text{Area} \approx 3.14 \times 49 = 153.86 \, \text{cm}^2 \]

### Step 6: Round the Area
Rounding 153.86 cm² gives approximately 154 cm².

### Final Answer
Thus, the area of the largest possible circle that can be inscribed in the rectangle EFGH is approximately **154 cm²**.

---