Single correct option
The line 3x + 4y = 24 forms a triangle with the coordinate axes. What is the distance from the origin to the centre of the circle inscribed in this triangle?
- A. 2
- B. √5
- C. 3
- D. 2√2
Show answer and solution
Answer: (D) 2√2
Intercepts: x-intercept = 24/3 = 8, y-intercept = 24/4 = 6. The triangle has vertices (0, 0), (8, 0), (0, 6) with a right angle at the origin; hypotenuse = √(64 + 36) = 10.
Inradius r = (8 + 6 − 10)/2 = 2.
The incircle touches both axes, so its centre is at (r, r) = (2, 2). Distance from the origin = √(2² + 2²) = 2√2.
Why others fail: 2 is the inradius itself (distance to each axis, not to the origin); √5 and 3 are slips.
Tip: For a right triangle at the origin with legs on the axes, the incentre is always (r, r).
Inradius r = (8 + 6 − 10)/2 = 2.
The incircle touches both axes, so its centre is at (r, r) = (2, 2). Distance from the origin = √(2² + 2²) = 2√2.
Why others fail: 2 is the inradius itself (distance to each axis, not to the origin); √5 and 3 are slips.
Tip: For a right triangle at the origin with legs on the axes, the incentre is always (r, r).
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