Question

In the given figure, \(\triangle ABC\) 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 any triangle, the area can also be expressed as \(Area = r \times s\), where \(r\) is inradius and \(s\) is semi-perimeter. This is another great way to find the radius!

Answer 1

Step 1: Understanding the Concept:
For a right-angled triangle, the radius of the incircle (inradius) can be calculated using the sides of the triangle. The tangents from a point to a circle are equal.
Step 2: Key Formula or Approach:
For a right triangle, Inradius \(r = \frac{P + B - H}{2}\).
Step 3: Detailed Explanation:
1. Given \(AB (P) = 4\) cm, \(BC (B) = 3\) cm. 2. Find Hypotenuse \(AC (H)\): \(\sqrt{4^2 + 3^2} = \sqrt{25} = 5\) cm. 3. Use the formula for inradius: \[ r = \frac{4 + 3 - 5}{2} \] \[ r = \frac{2}{2} = 1 \text{ cm} \]
Step 4: Final Answer:
The radius of the inscribed circle is 1 cm.