Problems from AMC12 (geometry)
1. (1986-3) ∠ACB = 90◦ , ∠BAC = 20◦ . If BD is the interior bisector of ∠ABC (D is
on AC), then ∠BDC =
(A) 40◦
(B) 45◦
(C) 50◦
(D) 55◦
(E) 60◦
2. (1984-4) A rectangle intersects a circle as shown: AB = 4, BC = 5, and DE = 3.
Then EF equals:
(D) 8
(E) 9
(A) 6
(B) 7
(C) 20
3
3. (1992-5) In the arrow-shaped polygon [see figure], the angles at vertices A, C, D, E
and F are right angles, BC = F G = 5, CD = F E = 20, DE = 10 and AB = AG. The area
of the polygon is closest to
(A) 288
(B) 291
(C) 294
(D) 297
(E) 300
4. (1999-7) What is the largest number of acute angles that a convex hexagon can have?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
5. (2004-12) Let A = (0, 9) and B = (0, 12). Points A0 and B 0 are on the line y = x, and
AA0 and BB 0 intersect
of A0 B 0 ?
√ at C = (2, 8). What is
√ the length √
(A) 2
(B) 2 2
(C) 3
(D) 2 + 2
(E) 3 2
6. (2006-13) The vertices of a 3 − 4 − 5 right triangle are the centers of three mutually
externally tangent circles, as shown. What is the sum of the areas of the three circles?
(C) 13π
(D) 27π
(E) 14π
(A) 12π
(B) 25π
2
2
7. (1999-16) What is the radius of a circle inscribed in a rhombus with diagonals of
length 10 and 24?
(A) 4
(B) 58/13
(C) 60/13
(D) 5
(E) 6
8. (1999-21) A circle is circumscribed about a triangle with sides 20, 21, and 29, thus
dividing the interior of the circle into four regions. Let A, B, and C be the areas of the
non-triangular regions, with C being the largest. Then
(A) A + B = C
(B) A + B + 210 = C
(C) A2 + B 2 = C 2
(D) 20A + 21B = 29C
(E) A12 + B12 = C12
9. (2006-23) Isosceles 4ABC has a right angle at C. Point P is inside
such
p 4ABC,
√
that P A = 11, P B = 7, and P C = 6. Legs AC and BC have length s = a + b 2, where
a and b are positive integers. What is a + b?
(A) 85
(B) 91
(C) 108
(D) 121
(E) 127