Geometry Chapter 8: Right Triangles and Trigonometry

Questions and Answers

What is the mean proportional of two positive numbers a and b?

The number x such that a/x = x/b.

What does the Altitude of Right Triangle Theorem state?

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.

What does the Right Triangle Geometric Mean Altitude Theorem state?

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments and its length is the geometric mean between the lengths of these two segments.

What does the Right Triangle Geometric Mean Leg Theorem state?

The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments, and the length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the adjacent segment. Signup and view all the answers

What is the Pythagorean Theorem?

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Signup and view all the answers

What is a Pythagorean Triple?

A set of three nonzero whole numbers that make the Pythagorean Theorem true. Signup and view all the answers

What does the Converse of the Pythagorean Theorem state?

If the sum of the squares of the lengths of the shortest sides of a triangle is equal to the square of the length of the longest side, then the triangle is a right triangle. Signup and view all the answers

What does Pythagorean Inequality Theorem 1 state?

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. Signup and view all the answers

What does Pythagorean Inequality Theorem 2 state?

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle. Signup and view all the answers

What does the 45°- 45°- 90° Triangle Theorem state?

In a 45°- 45°- 90° triangle, the legs of the triangle are congruent and the length of the hypotenuse is √2 times the length of a leg. Signup and view all the answers

What does the 30°- 60°- 90° Triangle Theorem state?

In a 30°- 60°- 90° triangle, the length of the hypotenuse is 2 times the length of the shorter leg and the length of the longer leg is √3 times the length of the shorter leg. Signup and view all the answers

What is a trig ratio?

A ratio of the lengths of two sides in a right triangle: sine, cosine, and tangent. Signup and view all the answers

What is the definition of sine?

sin x = opposite/hypotenuse Signup and view all the answers

What is the definition of cosine?

cos x = adjacent/hypotenuse Signup and view all the answers

What is the definition of tangent?

tan x = opposite/adjacent Signup and view all the answers

What is the 'trick' to remember basic trig ratios?

SOHCAHTOA Signup and view all the answers

What is the Inverse Sine of x?

The angle whose sine is equal to x. Signup and view all the answers

What is the Inverse Cosine of x?

The angle whose cosine is equal to x. Signup and view all the answers

What is the Inverse Tangent of x?

The angle whose tangent is equal to x. Signup and view all the answers

What is the angle of elevation?

The angle between the line of sight and the horizontal when an observer looks upward. Signup and view all the answers

What is the angle of depression?

The angle between the line of sight and the horizontal when an observer looks downward. Signup and view all the answers

Study Notes

Mean Proportional or Geometric Mean

Mean proportional between two positive numbers ( a ) and ( b ) is defined as ( x ) such that ( \frac{a}{x} = \frac{x}{b} ).
Solving for ( x ) gives ( x = \sqrt{ab} ).

Altitude of Right Triangle Theorem

Drawing an altitude to the hypotenuse of a right triangle creates two smaller triangles.
These smaller triangles are similar to both the original triangle and each other.

Right Triangle Geometric Mean Altitude Theorem

The altitude to the hypotenuse divides it into two segments.
The length of the altitude is the geometric mean of the two segment lengths.

Right Triangle Geometric Mean Leg Theorem

The altitude to the hypotenuse also defines the length of a triangle's leg.
A leg's length serves as the geometric mean between the hypotenuse length and the adjacent segment's length.

Pythagorean Theorem

Relates the lengths of the sides in a right triangle: ( a^2 + b^2 = c^2 ), where ( c ) is the hypotenuse.

Pythagorean Triple

A set of three positive integers ( (a, b, c) ) that satisfy the Pythagorean Theorem.

Converse of the Pythagorean Theorem

Establishes that if ( a^2 + b^2 = c^2 ), then the triangle is a right triangle.

Pythagorean Inequality Theorem 1

If ( c^2 < a^2 + b^2 ), the triangle is classified as acute.

Pythagorean Inequality Theorem 2

If ( c^2 > a^2 + b^2 ), the triangle is classified as obtuse.

45°- 45°- 90° Triangle Theorem

In a triangle with angles of 45°- 45°- 90°, the legs are congruent.
The hypotenuse measures ( \sqrt{2} ) times the length of a leg.

30°- 60°- 90° Triangle Theorem

For this triangle, the hypotenuse is twice the shorter leg's length.
The longer leg is ( \sqrt{3} ) times the length of the shorter leg.

Trigonometric Ratios

Sine, cosine, and tangent are ratios of the sides of a right triangle.

Sine

Defined as ( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} ).

Cosine

Defined as ( \cos x = \frac{\text{adjacent}}{\text{hypotenuse}} ).

Tangent

Defined as ( \tan x = \frac{\text{opposite}}{\text{adjacent}} ).

Mnemonic for Basic Trig Ratios

"SOHCAHTOA" serves as a memory aid for remembering sine, cosine, and tangent definitions.

Inverse Sine of x

Represents the angle whose sine value is ( x ).

Inverse Cosine of x

Represents the angle whose cosine value is ( x ).

Inverse Tangent of x

Represents the angle whose tangent value is ( x ).

Angle of Elevation

The angle formed by the line of sight looking upwards from an observer to an object.

Angle of Depression

The angle formed by the line of sight looking downwards from an observer to an object.

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Explore essential concepts of right triangles and trigonometry with flashcards. This quiz will help you understand the mean proportional and the Altitude of Right Triangle Theorem, enhancing your grasp of important geometric relationships.