Maths · Area of triangles using determinants
OPQR is a square and are the middle points of the sides and respectively, then t
OPQR is a square and \( M, N \) are the middle points of the sides \( P Q \) and \( Q R \) respectively, then the ratio of the areas of the square and the \( \triangle O M N \) is
- A. 4: 1
- B. 2: 1
- C. 8: 3
- D. 7: 3
Step-by-step solution
Let side length of square be a. Set O(0,0), P(a,0), Q(a,a), R(0,a). Then M is midpoint of PQ: M(a, a/2). N is midpoint of QR: N(a/2, a). Area of triangle OMN = 1/2 |0*(a/2 - a) + a*(a - 0) + (a/2)*(0 - a/2)| = 1/2 |a^2 - a^2/4| = 3a^2/8. Area of square = a^2. Ratio square:triangle = a^2 : (3a^2/8) = 8:3.
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