How to Master the Pythagorean Theorem and Right Triangles

The Pythagorean theorem states that in a right triangle the squares of the two legs add to the square of the hypotenuse. Its converse works backwards: if three lengths satisfy the equation, the triangle must be right.

• 45-45-90 triangle: This is an isosceles right triangle where the two legs are congruent, and the hypotenuse is (sqrt{2}) times the length of one leg.
• 30-60-90 triangle: In this triangle, the sides are in the ratio of (1: sqrt{3}: 2), with the smallest side opposite the (30^circ) angle and the longest side being the hypotenuse.

Examples

Practice Questions:

1. In a right triangle, if one leg measures (9 text{ cm} ) and the hypotenuse measures (15 text{ cm} ), find the length of the other leg.
2. Calculate the longer leg in a 30-60-90 triangle if the shorter leg (opposite the (30^circ) angle) measures (4 text{ cm} ).

1. Using the Pythagorean theorem, ( b^2 = 15^2 – 9^2 = 144 ) so ( b = 12 text{ cm} ).
2. The longer leg (opposite the (60^circ) angle) is (sqrt{3}) times the shorter leg, so it measures (4 times sqrt{3} approx 6.93 text{ cm} ).

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