Question

In triangle pqr, a is the point of intersection of all the altitudes and b is the point of intersection of all the angle bisectors of the triangle. If pbr = 105, then what is the value of par (in degrees

Answer 1

Answer:

150°

Step-by-step explanation:

In triangle pqr, a is the point of intersection of all the altitudes and b is the point of intersection of all the angle bisectors of the triangle. If pbr = 105°

in Δ pqr

b is the point of intersection of all the angle bisectors of the triangle

in Δ PBR

∠R / 2 +  ∠P /2 + ∠PBR  = 180°

∠PBR = 105°

=>∠R / 2 +  ∠P /2 + 105°  = 180°

=>∠R / 2 +  ∠P /2  = 75°

=> ∠R +  ∠P   = 150°

in Δ pqr

∠R +  ∠P + ∠Q = 180°

=> 150° + ∠Q = 180°

=> ∠Q = 30°

Now a is the point of intersection of all the altitudes

∠QRA + 90° + ∠Q = 180°

=> ∠QRA + 90° + 30° = 180°

=> ∠QRA = 60°

∠QPA + 90° + ∠Q = 180°

=> ∠QPA + 90° + 30° = 180°

=> ∠QPA = 60°

in ΔPAR

∠PAR + ∠APR  + ∠ARP = 180°

as

∠APR  = ∠P - ∠QPA = ∠P - 60°

∠ARP = ∠R - ∠QRA = ∠R - 60°

using these

∠PAR +  ∠P - 60° + ∠R - 60° = 180°

=>∠PAR + (∠P + ∠R) = 300°

Using ∠R +  ∠P   = 150°

=> ∠PAR + 150° = 300°

=> ∠PAR = 150°