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Geometry: Inscribed Circle in Square Arcs

XAT 2011 · Quantitative Ability & Data Interpretation · Medium · Geometry

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Consider a square ABCD of side 60 cm. It contains arcs BD and AC drawn with centres at A and D respectively. A circle is drawn such that it is tangent to side AB, and the arcs BD and AC. What is the radius of the circle?

  1. A.

    9 cm

  2. B.

    10 cm

  3. C.

    12 cm

  4. D.

    15 cm

  5. E.

    None of the above

Answer

B

Explanation

Let radius of circle be rr. Center of circle OO is at distance rr from ABAB. OD=60+rOD = 60 + r. QD=60−rQD = 60 - r. In right ΔDOQ\Delta DOQ, (60+r)2=(60−r)2+OQ2(60 + r)^2 = (60 - r)^2 + OQ^2. From ΔAOQ\Delta AOQ, AO=60−r,AQ=r  ⟹  OQ2=(60−r)2−r2=3600−120rAO = 60 - r, AQ = r \implies OQ^2 = (60 - r)^2 - r^2 = 3600 - 120r. Thus (60+r)2=(60−r)2+3600−120r  ⟹  240r=3600−120r  ⟹  360r=3600  ⟹  r=10(60+r)^2 = (60-r)^2 + 3600 - 120r \implies 240r = 3600 - 120r \implies 360r = 3600 \implies r = 10 cm.

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