document.write( "Question 1156829: Show that the lines y = 2x − 5 and −2x + 11y = 25 create chords of equal length when they intersect the circle x2 + y2 = 25. Make a large diagram, and measure the inscribed angle formed by these chords. Describe two ways of calculating its size to the nearest 0.1 degree. What is the angular size of the arc that is intercepted by this inscribed angle?
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Algebra.Com's Answer #779654 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "A good problem...

\n" ); document.write( "... from which you will learn nothing if we show you the complete solution.

\n" ); document.write( "So I'll give you some ideas and let you have the pleasure of solving the problem yourself.

\n" ); document.write( "The problem says the two chords of equal length form an inscribed angle. You can use that to check your work.

\n" ); document.write( "Solve the system of equations consisting of the circle and the first line; and similarly solve the system of the circle and the second line. You should find one point is a solution to both systems of equations, showing that the two lines intersect on the circle, forming an inscribed angle.

\n" ); document.write( "Having the coordinates of the points of intersection, use the distance formula to show that the lengths of the two chords are the same.

\n" ); document.write( "One of the points of intersection of the first line with the circle is (0,-5). You should be able to see that without doing any computations: the y-intercept of the first line is (0,-5), which is obviously a point on the circle.

\n" ); document.write( "Here is a graph:

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\n" ); document.write( "I see two ways to determine the measure of the inscribed angle.

\n" ); document.write( "First way:
\n" ); document.write( "Draw the radii of the circle that form the central angle corresponding to the inscribed angle.
\n" ); document.write( "The measure of the central angle is 90 degrees (the third quadrant), plus the measure of a small angle in the second quadrant; you can determine the measure of that angle from the coordinates of the point of intersection in the second quadrant.
\n" ); document.write( "Then of course the inscribed angle is half the central angle.

\n" ); document.write( "Second way:
\n" ); document.write( "Since the two chords are the same length, the bisector of the inscribed angle passes through the center of the circle.
\n" ); document.write( "So the measure of the inscribed angle is twice the measure of the angle determined by the origin and the vertex of the inscribed angle.

\n" ); document.write( "I strongly recommend going through the calculations for both methods; it is always gratifying to show that you can get the same answer by two very different methods.

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