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This essential trains for: SAT-I, GMAT, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.
Definition: any set of three positive integer numbers that satisfy the Pythagorean theorem.
If a right triangle has a hypotenuse of length x and legs of lengths y and z, then the values are related by:
Integer numbers that verify this equation can be found by solving it as a Diophantine equation. However, this is explained in a more advanced lesson.
For SAT purposes, it is sufficient to know that such triples of numbers exist, and to memorize the following ones:
Be able to quickly recognize scalings of them:
and so on.
The occurrence of two such numbers might be a clue to the solution.
The triangle is not right and two of these numbers are side lengths.
Note that, two sides of a triangle must share a vertex. Let us take two segments of lengths 3 and 4 that share an end: O. The side of length 3 can be almost any radius of the circle with radius 3 centered at O, while the side of length 4 can be almost any radius of the circle with radius 4 centered at O.
Therefore, there is an infinity of triangles with sides 3 and 4 and there is no guarantee that the third side will be 5 unless we also know that there is a 90° angle between the 3 and 4 sides.
Let's say there are two sides of lengths 3 and 4 and an angle of 90° but the angle is not between these sides, it is only adjacent to the shorter side:
The third side is a leg and has length:
Such 'false triples' may be used in problems in order to test the consistency of your knowledge.
Triangles that have side lengths forming a Pythagorean triple are not 'special triangles': 30°- 60°- 90° or 45°- 45°- 90°. This stands to reason since the ratios between side lengths in special triangles have irrational factors that can never be rationalized simultaneously for all three sides. Yet, students often think '3-4-5' triangles are also '30-60-90'.