Y. Davis Geometry Notes Chapter 8

Geometric mean The positive square root of the product of two numbers.

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.

Theorem 8.2 Geometric Mean (Altitude) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

Theorem 8.3 Geometric Mean (Leg) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into 2 segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Theorem 8.4 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. a²+b²=c²

Pythagorean Triple 3 whole numbers that satisfy the Pythagorean Theorem.

Theorem 8.5 Converse of the Pythagorean Theorem 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. a²+b²=c²

Theorem 8.6 Acute Triangle Theorem 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. a²+b²>c²

Theorem 8.7 Obtuse Triangle Theorem 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, hen the triangle is an obtuse triangle. a²+b²<c²

Theorem 8.8—45-45-90 Triangles In a 45-45-90 triangle, the legs l are congruent and the length of the hypotenuse h is times the length of a leg.

Theorem 8.9—30-60-90 Triangle In a 30-60-90 triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s, and the length of the longer leg l is times the length of the shorter leg.

Trigonometry The study of the measures of triangles.

Trigonometric Ratios Sine (sin) Cosine (cos) Tangent (tan)

Inverse Trigonometric Ratios

Angle of Elevation The angle formed by a horizontal line and an observer’s line of sight to an object above the horizontal line.

Angle of Depression The angle formed by a horizontal line and an observer’s line of sight to an object below the horizontal line.

Law of Sines Can be used to find side lengths and angle measures for any triangle.

Theorem 8.10—Law of Sines If , with lengths of a, b, and c, representing the lengths of the sides opposite the angles with measure A, B,and C, then

Theorem 8.11—Law of Cosines If , with lengths of a, b, and c, representing the lengths of the sides opposite the angles with measure A, B,and C, then

Vector A quantity that has both magnitude and direction.

Magnitude of a vector Size Length from initial point to terminal point.

Initial Point Starting point of the vector

Terminal point Ending point of a vector

Direction of a vector The angle it forms with the horizontal line. Or A measurement between 0 and 90 degrees east or west of the north-south line.

Resultant A single vector that is the sum of 2 or more vectors

Parallel vectors Have the same or opposite directions, but not necessarily the same magnitude

Opposite vectors Have the same magnitude, but opposite directions.

Equivalent vectors Have the same magnitude and direction.

Standard position When the initial point of a vector is the origin.

Component form

Vector Operations