# Geometry Theorems 8 Right Triangles

## 15 terms

### Theorem 8-1

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

### Theorem 8-3

If c² = a² + b², then m∠C = 90, and ∆ ABC is right.

### Theorem 8-4

If c² < a² + b², then m∠C < 90, and ∆ ABC is acute.

### Theorem 8-5

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

### Abbreviation

tan A = opposite / adjacent
sin A = opposite / hypotenuse
cos A = adjacent / hypotenuse

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