In the given figure, $\triangle ABC$ is a right triangle in which $\angle B = 90°$, AB = 4 cm and BC = 3 cm. Find the radius of the circle inscribed in the triangle ABC.
Generated by claude-sonnet-4-6 · 2026-06-15 10:25 · grounding rag
Model Answer
Step 1: Find the hypotenuse AC.
By Pythagoras theorem:
$$AC = \sqrt{AB^2 + BC^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \text{ cm}$$
Step 2: Use the incircle radius formula.
Let the incircle touch AB at P, BC at Q, and AC at R. Let the radius = r.
Since tangents from an external point are equal:
- From B: BP = BQ = r (tangent lengths from B, since ∠B = 90°)
- From A: AP = AR = AB − r = 4 − r
- From C: CQ = CR = BC − r = 3 − r
Since AR + CR = AC:
$$(4 - r) + (3 - r) = 5$$
$$7 - 2r = 5$$
$$2r = 2$$
$$\boxed{r = 1 \text{ cm}}$$
Source: Circles, Section 10.3
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Explanation
- Key property used: tangents from an external point are equal (Theorem 10.2).
- At the right angle vertex B, both tangent lengths equal r because the radius to the point of tangency is perpendicular to the side, forming a square of side r at B.
- The equation comes from AR + CR = AC (the two tangent segments from A and C add up to the hypotenuse).
- Always find AC first using Pythagoras — examiners expect this as a clear first step.