Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

diagram

$PQRS$ is a cyclic quadrilateral. If $\overline{SQ}$ bisects $\angle PQR$ prove chord $\overline{PS}$ = chord $\overline{SR}$.

share|improve this question
    
Sheesh, I already embedded the image for you and cleaned up your notation... –  J. M. Aug 4 '11 at 13:11
    
thanks, I'm inexperienced with the notation on web and as a new user I couldn't embed an image. –  MrK Aug 4 '11 at 13:13
    
Here's a sketch: $\angle PQS$ and $\angle RQS$ would be congruent, and thus subtend congruent arcs. Since $\stackrel{\frown}{PS}$ and $\stackrel{\frown}{SR}$ are congruent, then... –  J. M. Aug 4 '11 at 13:16

1 Answer 1

up vote 2 down vote accepted

Note that The Inscribed Angle Theorem says that $\angle PQS$ is half the central angle of chord $\overline{PS}$ and $\angle SQR$ is half the central angle of chord $\overline{SR}$. Since $\overline{SQ}$ bisects $\angle PQR$, $\angle PQS = \angle SQR$. Therefore, $\overline{PS}=\overline{SR}$ by SAS (side-angle-side).

share|improve this answer
    
thanks robjohn, I can't believe I overlooked that. I was trying to use a congruent triangles proof, but couldn't get it to work. –  MrK Aug 4 '11 at 13:29
    
out of curiosity is is possible to prove this using the triangles on the diagram? –  MrK Aug 4 '11 at 14:02
1  
@MrK: Without first proving a theorem that says that the length of a chord is dependent only on the diameter of the circle and the cosine of the inscribed angle subtended by the chord, I don't see a way without introducing some addition construction. –  robjohn Aug 4 '11 at 16:40

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.