Puzzle o' the Day 413!
Here are *four* medium-easy puzzles! You don't have to do them in order; pick one and solve it! While each of the puzzles can be solved in multiple ways, I recommend using the Intersecting Chords Theorem (or, in the case of the fourth puzzle, the Tangent-Secant Theorem; both theorems follow from the more general Power of a Point Theorem).
1. You have four points -- A,B,C,D. Segments AC and BD intersect at point E.
a. (Easy, assuming you remember the theorem, or clicked on the link above). If EA = 9, EC = 39, and EB = 13, what must ED be for the four points to be concyclic (i.e., lie on the same circle)?
b. (The puzzle: medium-easy). Assume A,B,C,D are concyclic, and that chords AC and BD are perpendicular. What's the radius of the circle?
c. (Follow-up: medium-easy). Generalize! Assume A,B,C,D are concyclic, and that chords AC and BD are perpendicular. If EA = a, EB = b, EC = c, ED = d, then what's the radius of the circle?
2. (Medium-easy). You have a circle. You draw a chord of length 16, which splits the circle into a minor arc (the small one) and a major arc (the large one). You can now draw a segment of length 2 connecting the midpoint of the minor arc to the midpoint of the chord. What's the radius of the circle?
3. (Medium-easy). You have a circle with center at point P. You draw a chord of length 22. You can now draw a segment of length 7 from P to the chord; this divides the chord into segments of length 10 and 12. What's the radius of the circle?
4. (Medium-easy). You have a circle, and a point X outside of it. You draw a line through X that's tangent to the circle; the distance from X to the point of tangency is 21. The distance from X to the closest point on the circle is 9. What's the radius of the circle?
1. You have four points -- A,B,C,D. Segments AC and BD intersect at point E.
a. (Easy, assuming you remember the theorem, or clicked on the link above). If EA = 9, EC = 39, and EB = 13, what must ED be for the four points to be concyclic (i.e., lie on the same circle)?
b. (The puzzle: medium-easy). Assume A,B,C,D are concyclic, and that chords AC and BD are perpendicular. What's the radius of the circle?
c. (Follow-up: medium-easy). Generalize! Assume A,B,C,D are concyclic, and that chords AC and BD are perpendicular. If EA = a, EB = b, EC = c, ED = d, then what's the radius of the circle?
2. (Medium-easy). You have a circle. You draw a chord of length 16, which splits the circle into a minor arc (the small one) and a major arc (the large one). You can now draw a segment of length 2 connecting the midpoint of the minor arc to the midpoint of the chord. What's the radius of the circle?
3. (Medium-easy). You have a circle with center at point P. You draw a chord of length 22. You can now draw a segment of length 7 from P to the chord; this divides the chord into segments of length 10 and 12. What's the radius of the circle?
4. (Medium-easy). You have a circle, and a point X outside of it. You draw a line through X that's tangent to the circle; the distance from X to the point of tangency is 21. The distance from X to the closest point on the circle is 9. What's the radius of the circle?