SOLUTION: Picture- Quadrilateral ABCD inscribed in a circle. If {{{m<D=75}}}, measure arc {{{AB = x^2}}}, measure arc {{{BC = 5x}}}, and measure arc {{{CD= 6x}}}, find x and {{{m<A}}}

Algebra ->  Circles -> SOLUTION: Picture- Quadrilateral ABCD inscribed in a circle. If {{{m<D=75}}}, measure arc {{{AB = x^2}}}, measure arc {{{BC = 5x}}}, and measure arc {{{CD= 6x}}}, find x and {{{m<A}}}      Log On


   

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
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Here's one way to draw out the starting diagram

I used GeoGebra to make the diagram.

Plot point E at the center of the circle.
Draw segments EA and EC to form angle AEC.

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.

This means angle AEC = 2*(inscribed angle D) = 2*75 = 150 degrees.

This further means that arc ABC is also 150 degrees.
The arc pieces AB = x^2 and BC = 5x add up to this 150 degree measure

(arc AB) + (arc BC) = arc ABC
x^2+5x = 150
x^2+5x-150 = 0

Use the quadratic formula to solve for x
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

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

x+=+%28-5%2B-sqrt%2825%2B600%29%29%2F%282%29

x+=+%28-5%2B-sqrt%28625%29%29%2F%282%29

x+=+%28-5%2B25%29%2F%282%29 or x+=+%28-5-25%29%2F%282%29

x+=+%2820%29%2F%282%29 or x+=+%28-30%29%2F%282%29

x+=+10 or x+=+-15
We'll ignore the negative x value because we cannot have negative angle measures.

Arc AB = x^2 = 10^2 = 100
Arc BC = 5x = 5*10 = 50
(arc AB) + (arc BC) = arc ABC
(100) + (50) = 150
150 = 150
So the x value checks out.

Now notice that inscribed angle A subtends the arc BCD
arc BCD = (arc BC) + (arc CD)
arc BCD = (5x) + (6x)
arc BCD = 11x
arc BCD = 11*10
arc BCD = 110 degrees

We cut that in half to find inscribed angle A (refer to the inscribed angle theorem)
angle A = (arc BCD)/2
angle A = (110)/2
angle A = 55 degrees

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Answer:
x = 10
Angle A = 55 degrees