2026 · Q65
A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question : ABC is a triangle inscribed in a semi-circle such that AC coincides with the diameter of the semi-circle. Let P be any point on the arc of the semi-circle. What is AP² + CP² + AC² equal to?
Statement IThe radius of the circle is 5 cm.
Statement IIAB = 6 cm and BC = 8 cm.
Which one of the following is correct in respect of the above Question and Statements?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- AThe Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
- BThe Question can be answered by using either Statement alone.
- CThe Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
- DThe Question can be answered even without using both the Statements.
Explanation
Since AC is a diameter, ∠APC = 90° for any P on the semicircle, so AP² + CP² = AC² and AP² + CP² + AC² = 2AC². Statement I gives AC = 10 (radius 5), and statement II gives AC = √(6² + 8²) = 10 (angle in a semicircle is a right angle); either alone yields 2·100 = 200. Either statement alone suffices, option B.