Harvard-MIT Mathematics Tournament March 15. 2003 Individual Round: Geometry Subject Test 1. AD and bC are both perpendicular to AB, and CD is perpendicular to AC. If AB=4 and BC=3. find CD B 2. As shown, U and C are points on the sides of triangle MNH such that MU =s UN=6, NC=20, CH=S, HM=25. If triangle UNC and quadrilateral MUCH have equal areas, what is s 3. A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12m. What is the area(in m") of the room? 4. Farmer John is inside of an ellipse with reflective sides, given by the equation x2/a2+ y / b2=l, with a>b>0. He is standing at the point (3, 0), and he shines a laser pointer in the y-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair(a, b)?Harvard-MIT Mathematics Tournament March 15, 2003 Individual Round: Geometry Subject Test 1. AD and BC are both perpendicular to AB, and CD is perpendicular to AC. If AB = 4 and BC = 3, find CD. C A D B 2. As shown, U and C are points on the sides of triangle MNH such that MU = s, UN = 6, NC = 20, CH = s, HM = 25. If triangle UNC and quadrilateral MUCH have equal areas, what is s? M N C H U 6 s 25 20 s 3. A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12m. What is the area (in m2 ) of the room? 4. Farmer John is inside of an ellipse with reflective sides, given by the equation x 2/a2 + y 2/b2 = 1, with a > b > 0. He is standing at the point (3, 0), and he shines a laser pointer in the y-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair (a, b)? 1