2016 AMC 10A Problems/Problem 24
Contents
- 1 Problem
- 2 Solution 1 (Algebra)
- 3 Solution 2 (Basic Algebra)
- 4 Solution 3 (Trigonometry Bash)
- 5 Solution 4 (Easier trig)
- 6 Solution 5 (Just Geometry)
- 7 Solution 6 (Ptolemy's Theorem)
- 8 Solution 7 (Trigonometry)
- 9 Solution 8 (Area By Brahmagupta's Formula)
- 10 Solution 9 (Cheap Solution - For when you are running out of time.)
- 11 Solution 10 (Measuring)
- 12 Solution 11 (Very Cheap Nonsensical Guessing)
- 13 See Also
Problem
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)
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 top 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,
Substituting values,
Finally, we multiply back the that we divided by at the beginning of the problem to get .
Solution 2 (Basic Algebra)
Let quadrilateral be inscribed in circle , where is the side of unknown length. Draw the radii from center to all four vertices of the quadrilateral, and draw the altitude of such that it passes through side at the point and meets side at the point .
By the Pythagorean Theorem, the length of is
Note that Let the length of be and the length of be ; then we have that
Furthermore,
Substituting this value of into the previous equation and evaluating for , we get:
The roots of this quadratic are found by using the quadratic formula:
If the length of is , then quadrilateral would be a square and thus, the radius of the circle would be Which is a contradiction. Therefore, our answer is
Solution 3 (Trigonometry Bash)
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 top of the circle). Now, draw the radii from center to and . Apply law of cosines on ; let . We get the following equation: Substituting the values in, we get Canceling out, we get Because , , and are congruent, . To find the remaining side (), we simply have to apply the law of cosines to . 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 4 (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 .
Solution 5 (Just Geometry)
Label AD intercept OB at E and OC at F.
From there, , thus:
because they are both radii of . Since , we have that . Similarly, .
and , so
Solution 6 (Ptolemy's Theorem)
Let . Let be the center of the circle. Then is twice the altitude of . Since is isosceles we can compute its area to be , hence .
Now by Ptolemy's Theorem we have This gives us:
Solution 7 (Trigonometry)
Since all three sides equal , they subtend three equal angles from the center. The right triangle between the center of the circle, a vertex, and the midpoint between two vertices has side lengths by the Pythagorean Theorem. Thus, the sine of half of the subtended angle is . Similarly, the cosine is . Since there are three sides, and since ,we seek to find . First, and by Pythagorean.
Solution 8 (Area By Brahmagupta's Formula)
For simplicity, scale everything down by a factor of 100. Let the inscribed trapezoid be , where and is the missing side length. Let . If and are the midpoints of and , respectively, the height of the trapezoid is . By the pythagorean theorem, and . Thus the height of the trapezoid is , so the area is . By Brahmagupta's formula, the area is . Setting these two equal, we get . Dividing both sides by and then squaring, we get . Expanding the right hand side and canceling the terms gives us . Rearranging and dividing by two, we get . Squaring both sides, we get . Rearranging, we get . Dividing by 4 we get . Factoring we get, , and since cannot be negative, we get . Since , . Scaling up by 100, we get .
Solution 9 (Cheap Solution - For when you are running out of time.)
WLOG, let , and let ABCD be inscribed in a clrcle with radius . We draw perpendiculars from and to , and label the intersections and , respectively. We can see that (because BCFE is a rectangle), and since is clearly greater than 200, and and since , which is part of segment , is an integer, than we conclude that is also an integer or of the form . There is no reason for to be of the form because it seems too arbitrary. The only other integer choice is .
Solution 10 (Measuring)
A quick aside -- this problem takes about 90 seconds to solve with an accurate compass and a good ruler. Draw the diagram, to scale, and measure the last side. It becomes very clear that the answer is . -- Jonathan Ko
Solution 11 (Very Cheap Nonsensical Guessing)
(While this is very quick, it is not recommended in general.) Note that this is only a good idea if you have backup by the means of some other method, and can be used to check your answer by logic. On high-numbered AMC problems, answers are generally integers. Therefore, we can readily omit (B), (C), and (D). We have either (A) or (E). If (A) was correct, then the quadrilateral would be a square, which would be too predictable, and thus highly unlikely for a final AMC problem. You could also quickly check that if the quadrilateral was a square, then the center of the circle would be the center of the square. This would mean that the diameter of the circle would be the diagonal length of the square, namely , and thus the radius would be , contradicting the problem constraints. The only answer left is
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2016 AMC 12A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 12 Problems and Solutions |
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