Brilliant Math & Science Wiki. Under classical principles, all participants are assumed to have the common knowledge of perfect rationality, meaning that every player is aware that other players are perfectly rational (and that they are aware that other players are aware that other players are rational, etc.).

However, this is not usually the case in practical settings, as equilibrium rarely occurs in actual play. In fact, the perfectly rational agent is often at a disadvantage, since they overestimate the depth of other players. For example, in the '2/3 of the average' game described in the previous section, classical principles would suggest that the perfectly rational agent would select the number 0. However, the winning number is usually much higher in practice. For example, 21.6 was the winning guess in a competition with over 19,000 participants [1], which is slightly below the number a level-2 thinker would select. Brilliant Math & Science Wiki. If where and are constants such that and which of the following could be the graph of ?

Solution: Tip: The parabola opens down if Tip: The parabola has a intercept at Because the parabola opens downward, and we can eliminate choices (A) and (B) where this is not the case. If we set , then . Hence, is the y-intercept. We are told that . In choice (D), and in choice (E), Therefore we can eliminate them. The graphs of and intersect at point where is a constant. Incorrect Choices: (A), (B), (C), and (D) Solution 2 eliminates these choices by plugging and checking. Divisibliity Rules (2,3,5,7,11,13,17,19,...) A positive integer is divisible by.

Brilliant Math & Science Wiki. Find the area of the triangle below.

Imgur Since the three side lengths are all equal to 6, the semiperimeter is . Relationship of side lengths and angles of a triangle. Relationship of sides to interior angles in a triangle In a triangle: The shortest side is always opposite the smallest interior angle The longest side is always opposite the largest interior angle Try this Drag the orange dots on the triangle below.

Recall that in a scalene triangle, all the sides have different lengths and all the interior angles have different measures. In such a triangle, the shortest side is always opposite the smallest angle. (These are shown in bold color above) Similarly, the longest side is opposite the largest angle. Brilliant Math & Science Wiki. Brilliant Math & Science Wiki. Pythagorean Triples. A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a2 + b2 = c2 Example: The smallest Pythagorean Triple is 3, 4 and 5.

Let's check it: Calculating this becomes: And that is true Triangles. Properties of Triangles Lightning Quiz Practice Problems Online. Area of a Right Triangle Formula. Right angled triangle is a special type of triangle.

One of it's internal angle is a right angle. Finding a Side in a Right-Angled Triangle. Find a Side when we know another Side and Angle We can find an unknown side in a right-angled triangle when we know:

Geometry - About Perimeter - Circumference of Quarter Circle - Mathematics Stack Exchange. The Law of Sines. The Law of Sines (or Sine Rule) is very useful for solving triangles: a sin A = b sin B = c sin C It works for any triangle: And it says that:

Brilliant Math & Science Wiki. Generating triples has always interested mathematicians and Euclid came up with a formula for generating Pythagorean triples.

Primitive Pythagorean triples are Pythagorean triples and such that and are coprime. Note however that this formula generates all primitive triples but not all non-primitive triples. Euclid gives us the following formula: where and are any two positive integers that are usually called parameters. Brilliant Math & Science Wiki. Evaluate exactly.

Solution: Unit Circle with angle greater than 2 pi Notice that the problem reduces to finding the value of in the above picture. Since , makes an angle of with the negative -axis. SAT Practice Test Results. Math FAQ: Distance, Rate, and Time. The formula Distance = Rate x Time expresses one of the most frequently used relations in algebra. Since an equation remains true as long as you divide through by the same non-zero element on each side, this formula can be written in different ways: To find rate, divide through on both sides by time: Distance Rate = ----------- Time Rate is distance (given in units such as miles, feet, kilometers, meters, etc.) divided by time (hours, minutes, seconds, etc.).

Rate can always be written as a fraction that has distance units in the numerator and time units in the denominator, e.g., 25 miles/1 hour. To find time, divide through on both sides by rate: Distance Time = ----------- Rate When using this equation, it's important to keep the units straight. Practice SAT Question Explanation. 10. In the figure above AD = 4, AB = 3 and CD = 9. What is the area of triangle AEC ? Correct Answer: D. Practice SAT Question Explanation.