The altitude is drawn to the hypotenuse of a right ∆, then the two ∆s formed are ∼ to the original ∆ and to each other.
Theorem 8-1 corollary 1
When the altitude is drawn to the hypotenuse of a right ∆, the length of the altitude is the geometric mean between the segments of the hypotenuse.
Theorem 8-1 corollary 2
When the altitude is drawn to the hypotenuse of a right ∆, each leg is the geometric mean between the hypotenuse and the segments of the hypotenuse that is adjacent to that leg.
Theorem 8-2 Pythagorean Theorem
In a right ∆, the square of the hypotenuse is equal to the sum of the squares of the legs.
Theorem 8-3 Converse of Pythagorean Theorem
If the square of one side of a ∆ is equal to the sum of the squares of the other two sides, then the ∆ is a right ∆.
Common Right Triangle Length
3,4,5 / 6,8,10 / 9,12,15 / 12,16,20 / 15,20,25
5,12,13 / 10, 24, 26
8, 15, 17
7, 24, 25
If c² = a² + b², then m∠C = 90, and ∆ ABC is right.
If c² < a² + b², then m∠C < 90, and ∆ ABC is acute.
If c² > a² + b², then m∠C > 90, and ∆ ABC is obtuse.
Theorem 8-6 45°-45°-90° Theorem
In a 45°-45°-90° ∆, the hypotenuse is √2 times as long as a leg.
Theorem 8-7 30°-60°-90° Theorem
In a 30°-60°-90° ∆, the hypetenuse is twices as long as the shorter let, and the longer leg is √3 as long as the shorter leg.
Trigonometry - Tangent Ratio - the ratio of the legs
tangent of ∠A = leg opposite ∠A / leg adjacent to ∠A
Trigonometry - Sine Ratio - the ratio that relate the opposite leg to the hypotenuse
sine of ∠A = leg opposite ∠A / hypotenuse
Trigonometry - Cosine Ratio - the ratio that relate the adjacent leg to the hypotenuse
cosine of ∠A = leg adjacent to ∠A / hypotenuse
tan A = opposite / adjacent
sin A = opposite / hypotenuse
cos A = adjacent / hypotenuse