Geometric Mean
the mean of two positive numbers a and b is the number x such that a/x=x/b. So, x^2 =ab and x=?ab
Similar Altitudes
if the altitude is drawn to the hypotnuse of a right triangle, then the two triangles formed are similar to the original triangle and to eachother
Geometric Mean (altitude) Theorem
The altitude drawn to the hypotenuse of aright triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments
Geometric Mean (Leg) Theorem
The altitudes drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg pf this triangle is the geometric mean between the length of the hypotenuse and the segment f the hypotenuse adjacent to that leg
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
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
Pythagorean Triples
If (a,b,c) it a triple, then any multiple (ax,bx,cx) is also a triple. (x must be posote
Pythagorean Triples ex.
3,4,5
5,12,13
8,15,17
7,24,25
Pythagorean Inequality Theorems (acute)
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 two sides, then the triangle is an acute triangle
Pythagorean Inequality Theorems (obtuse)
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
30, 60,90 Triangle Theorem
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 l is ?3 times the length of the shorter leg
45,45,90 Triangle theorem
In a 45,45,90 triangle, the legs l are congruent and the length of the hypotenuse h is ?2 times the length of the leg
Trigonometric ratio
a ratio of the length of the sides of a right triangle
Sine
angle= opposite side/hypotenuse side
Cosine
angle= adjacent side/hypotenuse side
Tangent
angle =opposite side/adjacent side
SohCahToa
anagram for the trigonometric ratios sin, cosine, and tangent
Inverse sine
If angle A is an acute angle and the sine of A is x, then the __________ _____ of x is the measure of angle A
Inverse cosine
If angle A is an acute angle and the cosine of A is x, then the _______ _____ of x is the measure of angle A
Inverse Tangent
If angle A is an acute angle and the tangent of A is x, then the _________ _________ of x is the measure of angle A
Angle of Elevation
the measurement of an angle from the ground going up
Angle of Depression
The measurement of an angle starting of the ground and going down to it
Solving a Triangle
finding the measures of all the missing angles and the lengths of all the missing sides of a triangle
The Law of Signs
ig triangle ABC has lengths a,b, and c, representing the lengths of the sides opposite the angles with measures A,B, and C, the sin A/a= Sin B/b = Sine C/c
Law of Cosines
If triangle ABC has lengths a,b, and c, representing the lengths of the sides opposite the angles with measures A,B, and C, then
a^2= b^2 + c^2- 2bc cos A
b^2= a^2+ c^2-2ac cos B
c^2= a^2+ b^2 -2ab cos C
Use with (SAS) or (SSS)