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October 2019 Junior Varsity Question 16

 
 
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October 2019 Junior Varsity Question 16
by Henry Yu - Wednesday, October 9, 2019, 5:36 PM
 

Question


In the diagram below, tangent AB¯¯¯¯¯¯¯¯AB¯ is perpendicular to diameter CD¯¯¯¯¯¯¯¯CD¯ at CCEE is the intersection of BD¯¯¯¯¯¯¯¯BD¯ with the circle.


EF¯¯¯¯¯¯¯¯EF¯ is a diameter with AAEEOO, and FF all collinear. If the measure of B∠B is 1212∘ more than the measure of A∠A, what is the measure of B∠B in degrees?


Solution


Let A∠A have measure xx and B∠B measure yy. Thus yx=12y−x=12.

EF¯¯¯¯¯¯¯¯EF¯ is a diameter, so arcs ECˆEC^ and CFˆCF^ add up to 180180∘. Therefore,

ECˆCFˆ2ECˆ(180ECˆ)ECˆ=A=2x=90+x.EC^−CF^2=∠AEC^−(180−EC^)=2xEC^=90+x.
Similarly,
CDˆCEˆ2180(90+x)90x=B=2y=2y.CD^−CE^2=∠B180−(90+x)=2y90−x=2y.
Combined with yx=12y−x=12 we can solve for x=22x=22 and y=34y=34. Thus B∠B measures 3434∘.

The correct answer is: 34


Can someone please explain how to get 180 - (90 + x) = 2y?

Thanks!

 
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Re: October 2019 Junior Varsity Question 16
by Henry Yu - Wednesday, October 9, 2019, 6:26 PM
 

Nvm, I figured it out.