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:
- 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.
- Calculate the longer leg in a 30-60-90 triangle if the shorter leg (opposite the \(30^\circ\) angle) measures \(4 \text{ cm} \).
- Using the Pythagorean theorem, \( b^2 = 15^2 – 9^2 = 144 \) so \( b = 12 \text{ cm} \).
- 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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