We will prove that SR ≅ ST, given that SR and ST are tangent to ⊙ P.
To do so, we will use congruent triangles. Let's start by drawing PR, PT, and PS.
Having drawn these segments, we have two triangles, △ SRP and △ STP. From here, we will continue by showing that these triangles are congruent.
As we can see, PR and PT are the radii of ⊙ P, and R and T are their corresponding points of tangency. Let's consider the Tangent Line to Circle Theorem.
Tangent Line to Circle Theorem
In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.
With this theorem, we can say that ∠ PRS and ∠ PTS are right angles.
Therefore, we can conclude that △ SRP and △ STP are right triangles. Next, recall that radii of a circle are congruent. Therefore, PR and PT are congruent segments.
Combining all this information, we can say that the hypotenuse and one leg of △ SRP are congruent to the hypotenuse and one leg of △ STP. Therefore, by the Hypotenuse-Leg Theorem, we have that △ SRP and △ STP are congruent triangles.
△ SRP ≅ △ STP
Finally, because corresponding parts of congruent triangles are congruent, we can conclude that SR≅ ST.
Let's summarize the above process in a flow proof.
