Consider a right angled triangle with in-radius is 2 cm and circum radius of 7 cm. What is the area of triangle?

1. ### Break Down the Problem

To find the area of a right-angled triangle when given the in-radius $r$ and circum-radius $R$, we can utilize the relationships between the sides of the triangle and these radii. In particular, the area $A$ can be determined using the formula:

$A = r \cdot s$

where $s$ is the semi-perimeter of the triangle, and also via the relationship with the circum-radius:

$A = \frac{abc}{4R}$

where $a$, $b$, and $c$ are the sides of the triangle.

2. ### Relevant Concepts

For a right-angled triangle:

• The circum-radius $R$ is given by $R = \frac{c}{2}$, where $c$ is the hypotenuse.
• The relationship between the in-radius $r$ and the area $A$ is given by $A = r \cdot s$, where $s$ is also expressed in terms of the sides.

3. ### Analysis and Detail

1. Calculate the semi-perimeter $s$: The semi-perimeter $s$ can be expressed in terms of the sides $a$, $b$, and hypotenuse $c$: $s = \frac{a + b + c}{2}$

2. Using the in-radius to find the area: From the in-radius: $A = r \cdot s = 2 \cdot s$

3. Using the circum-radius: Since we know $R = 7$, we can find $c$: $c = 2R = 2 \times 7 = 14$

4. Relationship between area, sides, and circum-radius: $A = \frac{abc}{4R} = \frac{abc}{28}$

5. Relating $a$ and $b$ with $c$: In the right-angled triangle, $c^2 = a^2 + b^2$. Set $c = 14$: $196 = a^2 + b^2$

6. Find $s$ and relate it to the area: Combining the area equations, $s = \frac{a + b + 14}{2}$, we balance the two equations for area.

4. ### Verify and Summarize

To find a solution for sides $a$ and $b$: We express $b$ in terms of $a$ using $b = \sqrt{196 - a^2}$ and substitute back to find a more direct relation with $s$.

However, we can directly solve for $A$:

1. $s = \frac{A + 14}{2r} = \frac{28 + 14}{2 \cdot 2} = \frac{28}{4} = 7$.

2. Area: From $A = 2 \cdot s$: $A = 2 \cdot 7 = 14 \text{ cm}^2$

Final Answer

The area of the triangle is $14 \text{ cm}^2$.