CDS 2026 (I) Elementary Mathematics, question 53

Topic: Geometry

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Q53

In a quadrilateral ABCD, the diagonals intersect at O. Let the area of the triangle ABD be pp. If AO:OC=m:nAO:OC=m:n, then what is the area of the triangle BCD ?

  1. (a) npm\frac{np}{m}Correct answer
  2. (b) mpn\frac{mp}{n}
  3. (c) n2pm2\frac{n^{2}p}{m^{2}}
  4. (d) m2pn2\frac{m^{2}p}{n^{2}}

Solution.

Answer: (a) npm\frac{np}{m}

  1. Triangles ABD and CBD have the same base BD.
  2. Their perpendicular heights to BD are in the ratio AO:OC=m:nAO:OC = m:n, because A, O and C lie on one line through BD.
  3. Their areas therefore have the same ratio m:n.
  4. Since area ABD=pABD = p, area BCD=np/mBCD = np/m.