Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
KVPY2018MathematicsProperties of Triangles

Let P   Q   R be an acute-angled triangle in which P   Q < Q   R   . From the vertex Q draw the altitude Q Q 1 the angle bisector Q Q 2 and the median Q Q 3 , with Q 1 , Q 2 , Q 3 lying on P   R . Then,

Options

  1. AP Q 1 < P Q 2 < P Q 3
  2. BP Q 2 < P Q 1 < P Q 3
  3. CP Q 1 < P Q 3 < P Q 2
  4. DP Q 3 < P Q 1 < P Q 2

Correct answer

A. P Q 1 < P Q 2 < P Q 3

Step-by-step solution

Given, P Q R is an acute angle triangle. P Q Q R ∠ Q R P ∠ Q P R P Q 3 = 1 2 P R P Q 2 : Q 2 R = r : p P Q 2 = r r + p P R But r p P Q 2 1 2 P R Comparison between altitude and angle bisector ∠ Q P Q 2 + ∠ P Q 2 Q + ∠ P Q Q 2 = ∠ R Q Q 2 + ∠ Q Q 2 R + ∠ Q R Q 2 ∴ ∠ P Q Q 2 = ∠ R Q Q 2 [since, Q Q 2 is angle bisector of ∠ Q ∠ Q P Q 2 + ∠ P Q 2 Q = ∠ Q Q 2 R + ∠ Q R Q 2 ∴ P Q Q R the ∠ Q P Q 2 ∠ Q R Q 2 Hence, ∠ Q Q 2 P ∠ Q Q 2 R But ∠ Q Q 2 P + ∠ Q Q 2 R = 180 ° Hence, ∠ Q Q 2 P 90 ° and ∠ Q Q 2 R > 90 ° ∵ Foot from

Practice Properties of Triangles on Quantrex Academy →

More from Properties of Triangles

In A B C , if A, B, C are in arithmetic progression then a ^2- ac + c ^2 ( ~A - C 2 )= 2025If in A B C, B=45^ , a=2( 3 +1) and area of A B C is 6+2 3 sq.units, then the side b = 2025In triangle A B C , if C=120^ , c= 19 and b=3 , then a= 2025In a A B C, 2 A+C=300^ . If the circumradius of the A B C is eight times its inradius then C 2 = 2025In A B C , if a=5, b=4 and (A-B)= 31 32 , then c= 2025In a triangle ABC, if A =30^ and b ( 3 +1)^2+2( 2 -1) = c ( 3 +1)^2-2( 2 -1) , then B= 2025In ABC if the line joining the circumcentre and the incentre is parallel to BC, then B + C = 2025In a triangle ABC, if r ₁: r ₂=3: 4 and r ₂: r ₃=2: 3 , then a : b : c = 2025 Full Properties of Triangles list All KVPY PYQs