In circle T, ∠PTQ ≅ ∠RTS. Circle T is shown. Line segments T P, T Q, T R, and T S are radii. Lines are drawn from point P to point Q and from point S to point R to form secants P Q and S R. Angles P T Q and S T R are congruent. The measure of arc S R is 66 degrees. The length of T R is 3 and the length of S R is 4. What is the length of PQ? 3 units 4 units 6 units 7 units

The length of PQ is 4 units ⇒ 2nd answer

Step-by-step explanation:

In circle T:

∵ TP , TQ , TR , TS are radii

- The radii of a circle are equal in length

∴ TP = TQ = TR = TS

∵ TR = 3 units

∴ TQ = 3 units

∴ TP = 3 units

∴ TS = 3 units

∵ ∠RTS is a central angle ⇒ its vertex is the center of the circle

∵ ∠RTS is subtended by arc SR

- The measure of the central angle is equal the measure of

   its subtended arc

∵ The measure of arc SR = 66°

∴ m∠RTS = 66°

∵ ∠PTQ ≅ ∠RTS

∴ m∠PTQ = 66°

In Δs STR and PTQ

∵ ST = PT ⇒ proved

∵ TR = TQ ⇒ proved

∵ m∠PTQ = m∠RTS

- By using SAS case of congruence theorem

∴ Δ STR ≅ Δ PTQ ⇒ SAS congruence theorem

- As a result of congruence

∴ SR ≅ PQ

∵ SR = 4 units

∴ PQ = 4 units

The length of PQ is 4 units

c.)

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