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This essential trains for: SAT-I, GMAT, SAT-II, AMC-8, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.

30°- 60°- 90° This triangle can be obtained by cutting an equilateral triangle along a line of symmetry.

If the length of the side of the equilateral triangle is 2 units, the line of symmetry cuts such a side in half - therefore, the length of the side opposite the 30° angle is 1 unit long.

From the Pythagorean theorem we get the length of the side that is opposite the 60° angle:

Since triangles that have the same angles are similar, all the triangles with this set of angles will have the lengths of their sides in the same ratios as the example:

Think of similar figures as scalings (shrinking, expanding) of a figure: like when enlarging a photograph. Then, if a side of a building is enlarged to double, all the other sides of the building are also enlarged to double.

Look at a set of various side length triples that are based on the 30°- 60°- 90° triangle. Try to find out what scale factor was used.

Typical 'trap' example: Constructing equivalent ratios using an irrational multiplicative factor:

45°- 45°- 90° This triangle can be obtained by cutting a square in half along a diagonal line of symmetry.

If the length of the side of the square is 1 unit, then the length of the diagonal is given by the Pythagorean theorem:

Since triangles that have the same angles are similar, all the triangles with this set of angles will have the lengths of their sides in the same ratios as the example:

Typical 'trap' example: Again, the examiner may want to trip us by using an irrational multiplicative factor:

or:

Another known trap is to set up a right triangle with two side lengths in a ratio of 2:1 but they are both legs. The student is led to believe it is a 30°- 60°- 90° triangle by the ratio of 1:2, not realizing that, in the case of the special triangle this is a ratio between one of the legs and the hypotenuse.