Final answer:
In a rectangle, the diagonals are congruent. We can use trigonometry to find the length of RP.
Step-by-step explanation:
In a rectangle, the diagonals are congruent, so if RP is one of the diagonals, then RQ is also equal to RP. Since RP and RQ form a right triangle, the angles RTS and RQT are complementary, which means their sum is 90 degrees (m∠RTS + m∠RQT = 90 degrees).
We are given that m∠PTS = 34 degrees. Since angles in the same segment of a circle are congruent, m∠RQT = m∠PTS = 34 degrees. Therefore, m∠RTS = 90 - 34 = 56 degrees.
Now we can use trigonometry to find the length of RP. In triangle RQT, we can use the sine function: sin(RTS) = RP / QS. Rearranging, RP = QS * sin(RTS). Substituting the values we know, RP = 10 * sin(56 degrees) ≈ 8.13 units. Therefore, RP is approximately 8.13 units.