2016 AMC 10A Problems/Problem 24
A quadrilateral is inscribed in a circle of radius . Three of the sides of this quadrilateral have length . What is the length of the fourth side?
Solution 1 (Algebra)
<Diagram and Reformatting Needed>
To save us from getting big numbers with lots of zeros behind them, let's divide all side lengths by for now, then multiply it back at the end of our solution.
Construct quadrilateral on the circle with being the missing side (Notice that since the side length is less than the radius, it will be very small on the bottom of the circle). Now, draw the radii from center to and . Let the intersection of and be point . Notice that and are perpendicular because is a kite.
We set lengths equal to . By the Pythagorean Theorem,
We solve for :
By Ptolemy's Theorem,
Finally, we multiply back the that we divided by at the beginning of the problem to get .
Solution 2 (Trigonometry Bash)
Construct quadrilateral on the circle with AD being the missing side (Notice that since the side length is less than the radius, it will be very small on the bottom of the circle). Now, draw the radii from center to and . Apply law of cosines on triangle with angle as . We get the following equation: Substituting the values in, we get Canceling out, we get To find the remaining side (), we simply have to apply law of cosines with angle on since the other triangles , , and are congruent. Now, to find , we can derive a formula that only uses : Plugging in , we get . Now, applying law of cosines on triangle , we get
Solution 3 (Easier trig)
Construct quadrilateral on the circle with being the missing side. Then, drop perpendiculars from and to (extended) line , and let these points be and , respectively. Also, let . From Law of Cosines on , we have . Now, since is isosceles with , we have that . By SSS congruence, we have that , so we have that , so . Thus, we have , so . Similarly, , and .
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