2026 · Q83
For the next three (03) items that follow : The centre O of a circle inside a triangle ABC is at a distance of 13 cm from each of the vertices of the triangle. The diameter of the circle is 10 cm and the circle touches only two sides of the triangle, AB and AC. Consider the following statements :
- I∠ABC lies between 60° and 90°.
- IIIf z is the distance in cm from the centre O of the circle to the midpoint of BC, then 7 cm < z < 8 cm.
Which of the statements given above is/are correct?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Explanation
Statement I: B = arccos(5/13) ≈ 67.4°, which lies between 60° and 90° — correct. Statement II: the distance from O to the midpoint of BC equals R·cos A = 13·cos A, where cos A = −cos 2B = 119/169, giving z = 13·119/169 = 119/13 ≈ 9.15 cm, which is not between 7 and 8 cm — statement II is false. Hence I only, option A.