Question

in circle t, ∠ptq ≅ ∠rts. what is the length of $overline{pq}$? 3 units 4 units 6 units 7 units

Explanation:

Step1: Recall circle - congruent - angle property

In a circle, if two central angles are congruent, then the chords they intercept are congruent. Since \(\angle PTQ\cong\angle RTS\), chord \(\overline{PQ}\) is congruent to chord \(\overline{RS}\).

Step2: Calculate the length of \(\overline{RS}\)

Using the distance formula for the length of a line - segment in a circle (or just by observing the given lengths in the right - triangle formed in the circle with segments related to \(\overline{RS}\)). If we consider the right - triangle with legs 3 and 4, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3\) and \(b = 4\), then \(c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\) is not relevant here. We can directly see from the figure that the length of \(\overline{RS}\) is composed of two segments of lengths 3 and 4. So, \(RS=3 + 4=7\).

Step3: Determine the length of \(\overline{PQ}\)

Since \(\overline{PQ}\cong\overline{RS}\), \(PQ = 7\) units.

Answer:

7 units