2017 AIME II Problems/Problem 3
A triangle has vertices , , and . The probability that a randomly chosen point inside the triangle is closer to vertex than to either vertex or vertex can be written as , where and are relatively prime positive integers. Find .
The set of all points closer to point than to point lie to the right of the perpendicular bisector of (line in the diagram), and the set of all points closer to point than to point lie below the perpendicular bisector of (line in the diagram). Therefore, the set of points inside the triangle that are closer to than to either vertex or vertex is bounded by quadrilateral . Because is the midpoint of and is the midpoint of , and . The coordinates of point is the solution to the system of equations defined by lines and . Using the point-slope form of a linear equation and the fact that the slope of the line perpendicular to a line with slope is , the equation for line is and the equation for line is . The solution of this system is . Using the shoelace formula on quadrilateral and triangle , the area of quadrilateral is and the area of triangle is . Finally, the probability that a randomly chosen point inside the triangle is closer to vertex than to vertex or vertex is the ratio of the area of quadrilateral to the area of , which is . The answer is .
Since we know the coordinates of all three vertices of the triangle, we can find the side lengths: , , and . We notice that the point where the three distances are the same is the circumcenter - so we use one of the triangle area formulas to find the circumradius, since we know what the area is. We rearrange to get We know that , and , so using the Pythagorean Theorem gives . This means . Similarly, we know that , and , so we get that , and so . Lastly, we know that , and , so we get that , and . Therefore, our answer is .
To start the problem, identify the two midpoints that connect and . This is because the midpoints of such lines is the mark at which the point will sway closer to vertex / or vertex . The midpoint of is , and the midpoint of is . Then, determine the line at which the distance between vertex and vertex are the same. Assuming that is the real value of and is the real value of , we can create a simple equation:
where the left side of the equation is for the distance to vertex and the right side of the equation is the distance to vertex .
Squaring both sides and then distributing, we get
Notice that , and thus there is no need to create another equation.
Simplifying, we get .
Divide both sides by 20, then simplify, and the line that represents equivalent distance between vertex and vertex is .
This line starts at the midpoint of , which is , and ends at the line , as represents equivalent distance between vertex and vertex . Plug in to the equation , and we get . Now that we have our four points that are , and , we can calculate the area of the quadrilateral in which a point is closer to vertex as opposed to either vertex or vertex . Simply draw a rectangle that has the points and , and then subtract the two triangles that appear in between.
Thus, the area of the quadrilateral is . Since the problem asks us for the probability that a point chosen inside the triangle is inside the quadrilateral, and because the area of is , the probability is , which means the final answer is .
Solution by IronicNinja~
Solution 4 (Shoelace Theorem/Formula)
Calculate the area of the triangle using the Shoelace Theorem on
Get the four points , and by any method from the above solutions. Then use the Shoelace Theorem to find the area of the region we want:
Therefore the probability is . Thus giving the final answer of .
Solution by phoenixfire
Draw the circumradii from the circumcenter to the three vertices. Drop perpendicular from the circumcenter to the sides. Note that since the triangle is isosceles, the perpendicular are in fact perpendicular bisectors. Therefore the region containing the points closer to B are in where is the circumcenter and points of contact of the perpendiculars and the sides. Therefore our fraction is . Our answer is then .
~ Prabh1512 (Edit by GeoWhiz4536)
Using the same graph and methods as Solution 1, find the coordinates of , , , and . Also, note that angles and are right angles, so is a cyclic quadrilateral. Then, use Brahmagupta's formula to determine the area of the quadrilateral, which is . Then find the area of triangle , which is . The probability is . Finally, we get our answer of
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