The length of a rectangle is 5 inches and its width is less than 4 inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is √6, find the width of the rectangle.
Algebra II / Trigonometry
Area Ratio, Circles and Square 1
Find the ratio of the area of the circle circumscribed about a given square to the area of the circle inscribed in that square.
Find Lengths of Two Sides
In △ABC, the measure of ∠A is 60°, the measure of ∠B is 45°, and the length of side AC is 8. What are the lengths of the other two sides? Do not use the Law of Sines! (You may use it afterwards to check.)
Show Angle is 30°
△ABC and CA = CB, ∠C = 20°.
Also, BD = DC and BE = BA.
Show that ∠BDE = 30°.
Hint: With center B and radius BD, construct an arc of a circle meeting BA (extended) at X and BC at Y. Draw the segments DX, DY; ponder.
Red/Blue Points, Equilateral Triangle
Suppose every point in a plane is colored either red or blue. Show that there is an equilateral triangle somewhere in the plane whose vertices are all the same color.
Red/Blue Points, Midpoint
Suppose every point in a plane is colored either red or blue. Show that there is some line segment such that its two end points and its midpoint are all the same color.
Robot Stones
To test a new robot, two hundred smooth white stones are placed 3 feet apart in a long straight line on the ground. A basket is placed at the beginning of the line, 3 feet away from the first stone. The robot starts at the basket and collects the stones one by one, putting each stone in the basket before getting the next stone. The robot never moves the basket. It takes awhile. How far does the robot travel? (What is the robot’s name?)
Collinear Points on Circle
In a rectangular coordinate plane, any circle that passes through (-2, -3) and
(2, 5) cannot also pass through (1989, y). Find the value of y.
One Sequence to Continue
What is the next term in this sequence: 2, 9, 28, 65, 126, 217, _____?
Collinear Vertices of Squares
If the three vertices A, B, and C in the three adjacent squares shown are collinear, find the value of x.