CHAPTER 8 Right Triangles

Similarity in Right Triangles SECTION 8-1 Similarity in Right Triangles

Geometric Mean If a, b, and x are positive numbers and a/x = x/b, then x is called the geometric mean between a and b and x = √ab

THEOREM 8 -1 If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other

Corollary 1 When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

Corollary 2 When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

The Pythagorean Theorem SECTION 8-2 The Pythagorean Theorem

THEOREM In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

The Converse of the Pythagorean Theorem SECTION 8-3 The Converse of the Pythagorean Theorem

Triangle In order to have a triangle, the sum of any two sides must be greater than the third side.

THEOREM If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Some Common Right Triangle Lengths 3,4,5 5,12,13 6,8,10 10,24,26 9,12,15 8,15,17 12,16,20 7,24,25 15,20,25

If c2 = a2 + b2, then mC = 90°, and ∆ABC is right THEOREM 8 - 3 If c2 = a2 + b2, then mC = 90°, and ∆ABC is right

If c2 < a2 + b2, then mC < 90°, and ∆ABC is acute THEOREM 8 - 4 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 - 5 If c2 > a2 + b2, then mC > 90°, and ∆ABC is obtuse

Special Right Triangles SECTION 8-4 Special Right Triangles

THEOREM 8 - 6 In a 45°- 45°- 90° triangle, the hypotenuse is √2 times as long as a leg.

THEOREM 8 - 7 In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

The Tangent and Cotangent Ratios SECTION 8-5 The Tangent and Cotangent Ratios

TRIGONOMETRIC RATIOS For any right triangle, there are six trigonometric ratios of the lengths of the sides of a triangle.

Ratio: length of side opposite A to the length of side adjacent to A Tangent A Ratio: length of side opposite A to the length of side adjacent to A

Ratio: length of side adjacent A to the length of side opposite to A Cotangent A Ratio: length of side adjacent A to the length of side opposite to A = 1/(tan A)

Examples Find tan M N 13 5 M P 12

The Sine, Cosecant, Cosine, and Secant Ratios SECTION 8-6 The Sine, Cosecant, Cosine, and Secant Ratios

Ratio: length of side opposite A to length of hypotenuse. Sine A Ratio: length of side opposite A to length of hypotenuse.

Examples Find sin M N 13 5 M P 12

Examples Find sin N N 13 5 M P 12

Ratio: length of hypotenuse of A to length of side opposite A. Cosecant A Ratio: length of hypotenuse of A to length of side opposite A. = 1/(sin A)

Ratio: Length of side adjacent A to the length of hypotenuse. Cosine A Ratio: Length of side adjacent A to the length of hypotenuse.

Examples Find cos M N 13 5 M P 12

Ratio: length of hypotenuse of A to length of side adjacent to A. Secant A Ratio: length of hypotenuse of A to length of side adjacent to A. = 1/(cos A)

Applications of Right Triangle Trigonometry SECTION 8-7 Applications of Right Triangle Trigonometry

The angle between the horizontal and the line of sight Angle of Depression The angle between the horizontal and the line of sight

The angle above the horizontal and the line of sight Angle of Elevation The angle above the horizontal and the line of sight

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