Question

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

Explanation:

Step1: Identify similar - triangles

Since $\angle PTQ\cong\angle RTS$, and vertical - angles are equal, and angles in the same segment of a circle are equal, $\triangle PTQ\sim\triangle RTS$.

Step2: Use the property of similar - triangles

For similar triangles $\triangle PTQ$ and $\triangle RTS$, the ratios of corresponding sides are equal. If we assume the ratio of similarity is based on the given side - lengths, and since the sides of $\triangle RTS$ are $3$ and $4$, and the corresponding sides of $\triangle PTQ$ will have the same ratio.
We know that if two triangles are similar, and the side lengths of one triangle are $a,b,c$ and the other are $a',b',c'$, then $\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}$. Here, if we assume the side corresponding to the side of length $3$ in $\triangle RTS$ and the side corresponding to the side of length $4$ in $\triangle RTS$ in $\triangle PTQ$, and by the property of similar triangles, if we consider the relationship between the sides of the two similar triangles formed by the intersecting chords in the circle, we can see that the side lengths of $\triangle PTQ$ will be in the same ratio as $\triangle RTS$.
The side lengths of $\triangle RTS$ are $3$ and $4$, so the side length of $\overline{PQ}$ corresponding to the sides of $\triangle RTS$ is such that if we assume the similarity relationship, and since the sides of the two similar triangles are in proportion, the length of $\overline{PQ}$ is $4$ units.

Answer:

4 units