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}}}

Question 1190631: Picture- Quadrilateral ABCD inscribed in a circle. If , measure arc , measure arc , and measure arc , find x and
Answer by math_tutor2020(3817) (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

Use a = 1, b = 5, c = -150

or

or

or
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

-----------------------------------

Answer:
x = 10
Angle A = 55 degrees

RELATED QUESTIONS
Picture- Quadrilateral ABCD inscribed in a circle. If m < D = 75*, measure arc AB = x^2,... (answered by math_tutor2020)

In the circle below, chord CD intersect chord AB at E http://i.imgur.com/4STGEkQ.jpg (answered by solver91311)

Hello, I would like some help completing this proof. The way my homework is set up is... (answered by mananth,ikleyn)

in circle O, AB=12 AC=13 and (answered by Alan3354)
In circle O, AX and BY intersect at E. If Angle BXY = 40°, Angle YBX = 55°, and... (answered by greenestamps)
Please help me with this problem? If arc AB, arc BC, and arc CD are non-overlapping... (answered by Theo)
If an angle is inscribed in a circle, then the measure of the angle equals one-half the... (answered by ikleyn)
What is the difference between an arc measure and an arc length in a... (answered by jontull)
What is the measure of minor arc BE? arc AB = 20 degrees, arc BC = 30 degrees, arc CD =... (answered by solver91311)