Download Chapter 9 Notes

Chapter 9 Notes
Section 9.1
• Theorem 9.1: 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 9.2: In a right triangle, the altitude from the right angle to the hypotenuse
divides the hypotenuse into two segments. The length of the altitude is the geometric
mean of the length of the two segments.
• Theorem 9.3: In a right triangle, the altitude from the right angle to the hypotenuse
divides the hypotenuse into two segments. The length of each leg of the right triangle
is the geometric mean of the lengths of the hypotenuse and the segment of the
hypotenuse this is adjacent to the leg.
Section 9.2
• Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the legs.
• Pythagorean Triple: is a set of three positive integers a, b, and c, that satisfy the
equation c2 = a2 + b2 (3, 4, and 5) for example.
Section 9.3
• Converse of the Pythagorean Theorem: If the square of the length of the longest
side of a triangle is equal to the sum of the squares of the other two sides, then the
triangle is a right triangle. If c2 = a2 + b2, then triangle ABC is a right triangle.
• Theorem 9.6: 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 acute.
• Theorem 9.7: 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 obtuse.
Section 9.4
• Special Right Triangles: Right triangles whose angle measures are 45 – 45 – 90
degrees, or 30 – 60 – 90 degrees, are called special right triangles.
• 45-45-90 Triangle Theorem: In a 45-45-90 degree triangle, the hypotenuse is 2
times as long as each leg.
• 30-60-90 Triangle Theorem: In a 30-60-90 degree triangle, the hypotenuse is twice
as long as the shorter leg, and the longer leg is
3 times as long as the shorter leg.
Section 9.5
•
Trigonometric Ratios: Let ABC be a right triangle. The sine (sin), the cosine (cos)
and the tangent (tan), of the acute angle  A , are defined as follows:
side opposite  A a

hypotenuse
c
side adjacent  A b
cos A 

hypotenuse
c
side opposite  A a
tan A 

side adjacent  A b
sin A =
Section 9.6
• To Solve a Right Triangle: means to determine the measures of all its angles and
sides.
The inverse sine of X is written sin -1 .
if the sin A  x, then the sin -1 x  m A
if the cos A  y, then the cos -1 y  m A
if the tan A  z , then the tan -1 z  m A
Section 9.7
• Vector: is a quantity that has both direction and magnitude (size), and is represented
as an arrow drawn between two points.
• Magnitude of a Vector AB : is the distance from the initial point (starting point) A to
the terminal point (ending point) B and written AB .
• Direction of a Vector: is determined by the angle it makes with a horizontal line. Two
vectors are equal if they have the same magnitude and direction. Two vectors are
parallel if they have the same or opposite directions.
• Component Form of a Vector: Combines the horizontal and vertical components.
The component form of PQ is X , Y , where x is the change in the x value,
and y is the change in the y value.
• The Sum of Two Vectors:
The sum of U  a1 , b1 and V  a2 , b2 is U  V  a1 +a2 , b1  b2 .