In a tangential quadrilateral PQRS, angle S is 90°. A circle with centre O and radius r touches the sides PQ, QR, RS and SP at points A, B, C and D respectively. If QR = 38 cm, RS = 25 cm and QA = 27 cm, what is the radius r of the circle?

Introduction / Context:
This is a higher level geometry question involving a tangential quadrilateral, which is a quadrilateral that has an incircle touching all four of its sides. Additionally, one of the angles is a right angle. The problem asks for the radius of the incircle based on side lengths and certain tangent segment lengths. It tests understanding of tangent properties and special behaviour at a right angle vertex.

Given Data / Assumptions:

- PQRS is a quadrilateral with an incircle touching all sides.
- Angle at vertex S is 90 degrees.
- The circle touches PQ, QR, RS and SP at A, B, C and D respectively.
- QR = 38 cm, RS = 25 cm and QA = 27 cm.
- Tangents drawn from the same external point to a circle have equal lengths.
- We need to find the radius r of the incircle.

Concept / Approach:
From the tangent property, segments from one vertex to the points of tangency on the adjacent sides are equal. For example, QA equals QB because both are tangents from Q. Similarly, RB equals RC, SC equals SD and PD equals PA. By using the given side lengths and these equalities, we can find the distances SC and SD along the sides meeting at the right angle S. In a right angle with an incircle touching both sides, the distances from the vertex along each side to the points of tangency are equal to the radius of the circle.

Step-by-Step Solution:
Step 1: From Q, tangents QA and QB are equal. Given QA = 27 cm, so QB = 27 cm.Step 2: On side QR, QB + BR = QR, so BR = QR - QB = 38 - 27 = 11 cm.Step 3: From R, tangents RB and RC are equal, so RC = RB = 11 cm.Step 4: On side RS, RC + CS = RS, so CS = RS - RC = 25 - 11 = 14 cm.Step 5: From S, tangents SC and SD are equal, so SD = SC = 14 cm.Step 6: At vertex S, the two sides RS and SP meet at a right angle, and the incircle touches RS at C and SP at D.Step 7: For a circle inscribed in a right angle, the distance from the vertex along each side to the tangency point equals the radius, so the radius r = SC = SD = 14 cm.

Verification / Alternative check:
Geometry of a circle in a right angle wedge shows that if the radius is r, then the tangency points on the two perpendicular sides are at distance r from the vertex. Here, since SC and SD are both 14 cm, they match this property. Also, the tangent segment calculations around the quadrilateral are consistent: QA and QB are 27 cm, RB and RC are 11 cm and SC and SD are 14 cm, which fit all side lengths given. This supports the correctness of r = 14 cm.

Why Other Options Are Wrong:
Values like 10 cm, 12 cm, 16 cm and 18 cm would make the distances SC and SD differ from the correctly computed tangent segments, breaking the equal tangent property from S. They would also not maintain consistency between QR, RS and the given QA value. These numbers are distractors that may result from incorrect subtraction or confusion about which segments equal the radius.

Common Pitfalls:
Many students do not exploit the tangent segment equalities from each vertex, leading to incorrect segment values. Others recognise the equalities but forget the special property of an incircle inside a right angle, where the distance from the vertex to each tangency point along the sides equals the radius. Some may also misread QA as PA instead of QA, which leads to wrong starting values. Reading carefully and using all tangent relationships is essential.

Final Answer:
Therefore, the radius of the circle is 14 cm.