Geometry

Chapter 8 Review

Geometric Mean

64 49 =8 7 =56

1 3 = 3

5 20 = 100 =10

abx=Find the geometric mean between the two numbers.

7. 5 and 20

8. 64 and 49

Corollary 1

piece of hypotenuse altitude altitude other piece

of hypotenuse

Y

A ZX

=

XA YA=

YA AZ

Corollary 2

hypotenuse leg leg piece of

hyp. adj. to leg

Y

A ZX

=

XZ XYFor legXY: =

XY XA

Corollary 2

Y

A ZX

XZ YZFor legYZ: =

YZ AZ

hypotenuse leg leg piece of

hyp. adj. to leg

=

Pythagorean TheoremPythagorean Theorem

In a right triangle, the square of the In a right triangle, the square of the hypotenuse is equal to the sum of hypotenuse is equal to the sum of the squares of the legs. the squares of the legs.

a

bc

C

A

B

222 cba

If the square of one side of a triangle If the square of one side of a triangle is equal to the sum of the squares of is equal to the sum of the squares of the other two sides, then the the other two sides, then the triangle is a right triangle.triangle is a right triangle.

Theorem: Converse of Theorem: Converse of the Pythagorean the Pythagorean

TheoremTheorem

a

bcIf c² = a² + b² Rt. ∆

If the square of the longest side of a If the square of the longest side of a triangle is triangle is greater thangreater than the sum of the sum of the squares of the other two sides, the squares of the other two sides, then the triangle is an then the triangle is an obtuseobtuse triangle.triangle.

TheoremTheorem

a

bcIf c² > a² + b² Obtuse ∆

obtuse

If the square of the longest side of a If the square of the longest side of a triangle is triangle is less thanless than the sum of the the sum of the squares of the other two sides, then squares of the other two sides, then the triangle is an the triangle is an acuteacute triangle. triangle.

TheoremTheorem

a

b

cIf c² < a² + b² Acute ∆

acute

45-45-90 Triangles

45o

45o

x

x2x

Since all 45-45-90 triangles are similar, by AA Similarity Postulate, this formula works for all 45-45-90 triangles.

The formula.

30-60-90 Triangles

60o

30o

x

2x

Since all 30-60-90 triangles are similar, by AA Similarity Postulate, this formula works for all 30-60-90 triangles.

The formula.

3x

Tangent ratio = adj.opp.

legadjacentlegopposite

Atan

oppositeleg

adjacent leg

hypotenuse

A

Tangent of <A:

The tangent ratio is the ratio of thelength of the legs in a Rt. ∆

Sine and Cosine Ratios

hypotenuselegopposite

Asin

oppositeleg

adjacent leg

hypotenuse

A

Sine of <A:

The sine ratio is the ratio of thelength of the legs in a Rt. ∆

hypotenuselega

Adjacent

cos

oppositeleg

adjacent leg

hypotenuse

A

Cosine of <A:

HW W.S. Let’s do the odds on Chapter 8 side

together!