document.write( "Question 1190631: Picture- Quadrilateral ABCD inscribed in a circle. If \"m%3CD=75\", measure arc \"AB+=+x%5E2\", measure arc \"BC+=+5x\", and measure arc \"CD=+6x\", find x and \"m%3CA\" \n" ); document.write( "
Algebra.Com's Answer #822337 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "Here's one way to draw out the starting diagram
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\n" ); document.write( "I used GeoGebra to make the diagram.\r
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\n" ); document.write( "\n" ); document.write( "Plot point E at the center of the circle.
\n" ); document.write( "Draw segments EA and EC to form angle AEC.\r
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\n" ); document.write( "\n" ); document.write( "By the inscribed angle theorem, the inscribed angle D = 75 is exactly half of the central angle AEC because both subtend the same arc ABC.\r
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\n" ); document.write( "\n" ); document.write( "This means angle AEC = 2*(inscribed angle D) = 2*75 = 150 degrees.\r
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\n" ); document.write( "\n" ); document.write( "This further means that arc ABC is also 150 degrees.
\n" ); document.write( "The arc pieces AB = x^2 and BC = 5x add up to this 150 degree measure\r
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\n" ); document.write( "\n" ); document.write( "(arc AB) + (arc BC) = arc ABC
\n" ); document.write( "x^2+5x = 150
\n" ); document.write( "x^2+5x-150 = 0\r
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\n" ); document.write( "\n" ); document.write( "Use the quadratic formula to solve for x
\n" ); document.write( "\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+%28-%285%29%2B-sqrt%28%285%29%5E2-4%281%29%28-150%29%29%29%2F%282%281%29%29\" Use a = 1, b = 5, c = -150\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+%28-5%2B-sqrt%2825%2B600%29%29%2F%282%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+%28-5%2B-sqrt%28625%29%29%2F%282%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+%28-5%2B25%29%2F%282%29\" or \"x+=+%28-5-25%29%2F%282%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+%2820%29%2F%282%29\" or \"x+=+%28-30%29%2F%282%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+10\" or \"x+=+-15\"
\n" ); document.write( "We'll ignore the negative x value because we cannot have negative angle measures.\r
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\n" ); document.write( "\n" ); document.write( "Arc AB = x^2 = 10^2 = 100
\n" ); document.write( "Arc BC = 5x = 5*10 = 50
\n" ); document.write( "(arc AB) + (arc BC) = arc ABC
\n" ); document.write( "(100) + (50) = 150
\n" ); document.write( "150 = 150
\n" ); document.write( "So the x value checks out.\r
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\n" ); document.write( "\n" ); document.write( "Now notice that inscribed angle A subtends the arc BCD
\n" ); document.write( "arc BCD = (arc BC) + (arc CD)
\n" ); document.write( "arc BCD = (5x) + (6x)
\n" ); document.write( "arc BCD = 11x
\n" ); document.write( "arc BCD = 11*10
\n" ); document.write( "arc BCD = 110 degrees\r
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\n" ); document.write( "\n" ); document.write( "We cut that in half to find inscribed angle A (refer to the inscribed angle theorem)
\n" ); document.write( "angle A = (arc BCD)/2
\n" ); document.write( "angle A = (110)/2
\n" ); document.write( "angle A = 55 degrees\r
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\n" ); document.write( "\n" ); document.write( "-----------------------------------\r
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\n" ); document.write( "\n" ); document.write( "Answer:
\n" ); document.write( "x = 10
\n" ); document.write( "Angle A = 55 degrees
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