Question

In the given figure, \(\Delta ABC\) is a right triangle in which \(\angle B = 90^\circ\), \(AB = 4\) cm and \(BC = 3\) cm. Find the radius of the circle inscribed in the triangle \(ABC\).

Hint:

For a right-angled triangle with sides \(a, b\) and hypotenuse \(c\), the inradius is always \(\frac{a+b-c}{2}\). This is much faster than using \(\frac{\text{Area}}{s}\) during competitive exams.

Answer 1

Step 1: Understanding the Concept:
The inradius (\(r\)) of any triangle can be calculated using the formula \(r = \frac{\text{Area}}{\text{Semi-perimeter}}\). For a right-angled triangle, there is a specific shortcut formula.
Step 2: Key Formula or Approach:
Shortcut for right triangle: \(r = \frac{P + B - H}{2}\) where \(P\) is perpendicular, \(B\) is base, and \(H\) is hypotenuse.
Step 3: Detailed Explanation:
In \(\Delta ABC\), \(AB = 4\) cm and \(BC = 3\) cm.
Using Pythagoras theorem to find hypotenuse \(AC\):
\[ AC = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \text{ cm} \]
Using the inradius formula for right triangle:
\[ r = \frac{AB + BC - AC}{2} \]
\[ r = \frac{4 + 3 - 5}{2} = \frac{2}{2} = 1 \text{ cm} \]
Step 4: Final Answer:
The radius of the inscribed circle is 1 cm.