Question

A student drew a circle and two secant segment. He concluded that if PQ ~= PS, Then QR ~= ST. Do you agree with the student’s conclusion? Why or why not?

Answer 1

Answer:

  Yes

Step-by-step explanation:

You can get there a couple of ways. One makes use of the secant rules that tell you ...

  PQ × PR = PS × PT

Substituting for PR and PT, you have ...

  PQ × (PQ + QR) = PS × (PS + ST)

  PQ² + PQ×QR = PS² + PS×ST

Substituting PQ for PS everywhere, we have ...

  PQ² + PQ×QR = PQ² + PQ×ST

Dividing by PQ gives ...

  PQ + QR = PQ + ST

and subtracting PQ leads us to the conclusion ...

  QR = ST

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Another way to look at it is to draw the chord QS. Then ΔQPS is an isosceles triangle, and the perpendicular bisector of QS bisects ∠P and also goes through the circle center. Then the figure is symmetrical about that diameter secant, making QR ≅ ST.