Circle T has diameters RP and QS. The measure of ∠RTQ is 12° less than the measure of ∠RTS.

Circle T is shown. Line segments T S, T R, T Q, and T P are radii. Lines are drawn to connect points S and R and points P and Q to form secants. Angles R T S and Q T P are congruent.

What is the measure of Arc Q P?

78°
84°
88°
96°

The measure of Arc Q P is 96°. We also know that ∠QTP is central angle, then the measure of arc QP is 96°.

Step-by-step explanation:

Step 1

If QS is a circle diameter,

then m∠QTS=180°.

Let x be the measure of angle RTQ: ∠RTQ =x.

so, let ∠RTQ = x

Step 2

According to the question,

∠RTQ = ∠RTS - 12°

⇒ ∠RTS = x + 12°

∴ ∠QTS = ∠RTQ + ∠RTS

= x + x + 12° = 2x + 12° = 180°

⇒ 2x = 168°

⇒ x = 84°

⇒ ∠RTQ = 84°

Step 3

Now,

∵∠QTP and ∠RTS are vertical angles

∴ ∠QTP = 84° + 12° = 96°

As ∠QTP is the central angle, hence the measure of arc QP is 96°

Step 4

The Measure of arc QP = 96°

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