Geometry Ch.8

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)