2007 AIME I Problems/Problem 13
A square pyramid with base and vertex has eight edges of length 4. A plane passes through the midpoints of , , and . The plane's intersection with the pyramid has an area that can be expressed as . Find .
Note first that the intersection is a pentagon.
Use 3D analytical geometry, setting the origin as the center of the square base and the pyramid’s points oriented as shown above. . Using the coordinates of the three points of intersection (), it is possible to determine the equation of the plane. The equation of a plane resembles , and using the points we find that , , and . It is then .
Write the equation of the lines and substitute to find that the other two points of intersection on , are . To find the area of the pentagon, break it up into pieces (an isosceles triangle on the top, an isosceles trapezoid on the bottom). Using the distance formula (), it is possible to find that the area of the triangle is . The trapezoid has area . In total, the area is , and the solution is .
|2007 AIME I (Problems • Answer Key • Resources)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15|
|All AIME Problems and Solutions|