Question

In \(\triangle ABC\), if \(\angle A = 90^\circ\), then \((r_2 - r_1)(r_3 - r_1) = \)

• A. \(r_2 r_3\)
• B. \(\mathbf{2 r_2 r_3}\)
• C. \(4 r_2 r_3\)
• D. \(2\)

Hint:

Remember standard inradius–exradius identities for right-angled triangles to solve such problems quickly.

Answer 1

The Correct Option is B

In any triangle, the **exradii** \(r_2, r_3\) (opposite to B and C), and inradius \(r_1\) are related by identity: If \(\angle A = 90^\circ\), then the relation: \[ (r_2 - r_1)(r_3 - r_1) = 2 r_2 r_3 \] This is a standard result derivable from exradius and inradius properties in right-angled triangles.