Problem of the Week: Summer (June-September) 1995
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As the academic year in the United States wound down in June of 1995, the Forum's Geometry Problem of the Week continued with a series taken from some math contest problem sets. We give you the problems here without the solutions so you can enjoy working through them at your leisure.
- June 12-16:
In triangle ABC, AC=18, and D is the point on AC for which AD=5. Perpendiculars drawn from D to AB and CB have lengths of 4 and 5 respectively. What is the area of triangle ABC?
- June 19-23:
In parallelogram ABCD, the bisector of angle <ABC intersects AD at P. If PD=5, BP=6, and CP=6, what is the value of AB?
- June 26-30:
One diagonal of a square serves as the shorter base of a trapezoid, and a line through one of the vertices of the square contains the other base. The legs of the trapezoid are extensions of two sides of the square. If the area of the square is 2800, what is the area of the trapezoid?
- July 3-7:
In an isosceles triangle, the perpendicular bisector of one leg passes through the midpoint of the base. If the length of this leg is 10, how long is the base?
- July 10-14:
Semicircles drawn on each side of a triangle have areas of 9pi, 16pi, and 25pi. What is the area of the triangle?
- July 17-21:
One altitude of an equilateral triangle is a side of one square, and one side of the same equilateral triangle is a side of a second sqaure. The area of the larger of these squares is 56. What is the area of the smaller of these squares?
- July 24-28:
Regular hexagon ABCDEF has side AF in common with regular hexagon AFGHIJ and side BC in common with regular hexagon BCKLMN. All three hexagons are coplanar and nonoverlapping. If AB = 64, what is the value of JN?
- July 31-August 4:
The lengths of the sides of a triangle are 25, 29, and 36. There is a point on the longest side of the triangle whose distance from the opposite vertex is 20. What is the distance from this point to the midpoint of the shortest side?
- August 7-11:
Both legs of an isosceles triangle are radii of a circle, and the length of each radius is 6. The distance from the center of the circle to a point P on the base of the triangle is 4. If the distances from P to the triangle's other vertices are 5 and x, what is the value of x?
- August 14-18:
Two vertices of an equilateral triangle lie on a diameter of a circle whose area is 36pi, and the third vertex lies on the circle. What is the largest possible area of the triangle?
- August 21-25:
In a trapezoid, the lengths of the bases are 4 and 16, and the lower base angles are 30 degrees and 60 degrees. What is the distance between the midpoints of the two bases?
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- August 28-September 1:
What is the degree-measure of the acute angle formed by extending sides AB and ED of regular nine-sided polygon ABCDEFGHI until these extensions meet?
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