Area
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?
54sqrt3 - 18pi
In the diagram below, the area of triangle EBD is one third of the area of triangle ABC. Segment DE is perpendicular to segment AB. What is BD?
(4sqrt3)/3
Let ABC be an isosceles triangle with BC = AC and <ACB = 40. Construct the circle with diameter BC, and let D and E be the other intersection points of the circle with the sides AC and AB, respectively. Let F be the intersection of the diagonals of the quadrilateral BCDE. What is the degree measure of <BFC?
110
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
(sqrt5 - 1)/2
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles?
3sqrt3 - pi
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point A falls on point B. What is the length in inches of the crease?
15/8
In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4:5. What is the degree measure of angle BCD?
130
Quadrilateral ABCD with side lengths AB = 7, BC = 24, CD = 20, DA = 15 is inscribed in a circle. What is the area inside the circle but outside the quadrilateral?
(625pi - 936)/4
In rectangle ABCD, AB = 1, BC = 2, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of GE. What is the area of the shaded region?
1/6
The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD = 3. Point E is chosen so that ED = 5 and line ED is perpendicular to line AD. Segment AE intersects the circle at a point C between A and E. What is the area of triangle ABC?
140/37
In triangle ABC, angles A and B measure 60 degrees and 45 degrees, respectively. The bisector of angle A intersects BC at T, and AT = 24. What is the area of triangle ABC?
72sqrt3 + 216
In triangle ABC, AB = 86, and AC = 97. A circle with center A and radius AB intersects BC at points B and X. Moreover BX and CX have integer lengths. What is BC?
61