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KVPY2018MathematicsProperties of Triangles

Consider a Δ   P Q R in which the relation Q R 2 + P R 2 = 5   P Q 2 holds. Let G be the point of intersection of medians P   M and Q   N   . Then, ∠   Q G M is always

Options

  1. Aless than 45 °
  2. Bobtuse
  3. CA right angle
  4. Dacute and larger than 45 °

Correct answer

C. A right angle

Step-by-step solution

In Δ P Q R Given, Q R 2 + P R 2 = 5 P Q 2 Median P M and Q N intersect of G . Q G = 2 3 Q N , G M = 1 3 P M ⇒ Q G 2 + G M 2 = 2 3 Q N 2 + 1 3 P M 2 = 4 9 Q N 2 + 1 9 P M 2 = 4 9 2 P Q 2 + 2 Q R 2 - P R 2 4 + 1 9 2 P Q 2 + 2 P R 2 - Q R 2 4 = 1 9 8 P Q 2 + 8 Q R 2 - 4 P R 2 + 2 P Q 2 + 2 P R 2 - Q R 2 4 = 1 9 10 P Q 2 + 7 Q R 2 - 2 P R 2 4 = 1 9 2 5 P Q 2 - P R 2 + 7 Q R 2 4 = 1 9 2 Q R 2 + 7 Q R 2 4 = 1 4 Q R 2 = Q M 2 ∴ O G 2 + G M 2 = Q M 2 ∴ ∠ Q G M = 90 °

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