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In the figure given below, the side QR of $Δ P Q R$ is produced to a point S. If the bisectors of $∠ P Q R$ and $∠ P R S$ meet at point T, then what is the relation between $∠ QTR$ and $∠ Q P R$ ?

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∠ QTR = 14 ∠ QPR

∠ QTR = 2 ∠ QPR

∠ QTR = 12 ∠ QPR

∠ QTR = 4 ∠ QPR

answer is C.

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Detailed Solution

Given data,
Side of QR of triangle PQR is produced to a point S.
The angle bisectors

meet at point T, therefore QT and RT are the angle bisectors of

Applying Exterior angle property in triangle PQR,

∴ ∠ P R S = ∠ R P Q + ∠ P Q R

⇒ ∠ T R S = 12 ∠ R P Q + ∠ T Q R ... (2) using eq . (1)

Applying Exterior angle property in triangle QRT,

Equating the equations (2) and (3), we get,

∠ T Q R + 12 ∠ Q P R = ∠ T Q R + ∠ Q T R ⇒ 12 ∠ Q P R = ∠ Q T R ∴ ∠ Q T R = 12 ∠ Q P R

Therefore, the relation between

∠ QTR

and

∠ Q P R

∠ Q T R = 12 ∠ Q P R .

Hence, option (3) is correct.

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