Chapter 9 - Geometry

Pythagorean Theorem

In a right triangle, c²=a²+b²

Converse of the Pythagorean Theorem

If c²=a²+b², then the triangle is a right triangle

Pythagorean Inequalities Theorem

For any triangle triangle ABC, where c is the length of the longest side

if c²<a²+b², then triangle is acute

if c²>a²+b², then triangle is obtuse

45°-45°-90° Triangle Theorem

in this type of triangle, hypotenuse is √2 times as long as each leg

30°-60°-90° Triangle Theorem

in this type of triangle,

• hypotenuse is twice as long as shorter leg

• longer leg is √3 times as long as shorter leg

Right Triangle Similarity Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle and to each other.

geometric mean

of 2 positive numbers a and b is the positive x that satisfies a/x=x/b. so, x²=ab, and x=√ab

geometric mean (Altitude) Theorem

length of altitude is the geometric mean of lengths of 2 segments of the hypotenuse

• e.g. CD²=ADxBD

geometric mean (Leg) Theorem

length of each leg of right triangle is geometric mean of the lengths of hypotenuse & segment of hypotenuse that is adjacent to the leg

trigonometric ratio

ratio of lengths of 2 sides in right triangle

tangent ratio

trigonometric ratio for acute angles involving lengths of legs of right triangle

round values of trigonometric ratios to

4 decimal places

round lengths to

nearest tenth

angle of elevation

angle that upward line of sight makes w/ a horizontal line

sin A =

cos (90°-A) = cos B

cos B =

sin (90°-B) = sin A

angle of depression

angle that a downward line of sight makes w/ a horizontal line

“tan^-1x” read as

“the inverse tangent of x”

Inverse trigonometric ratios (e.g. inverse tangent)

If tan A = x, then tan^-1x = m∠A.

Solving right triangle

find all unknown side lengths & ∠ measures

• must know either of the following:

  • 2 side lengths

  • 1 side length & measure of 1 acute ∠

Area of any triangle

½ the product of length of 2 sides times the sine of their included ∠

• e.g. Area = 1/2 ⋅ bc ⋅ sin A

Law of Sines cases

• 2 angles & length of any side - AAS, ASA

• angle opposite 1 of the 2 sides - SSA

Law of Sines

sin A/a=sin B/b=sin C/c and the other way around

Law of Cosines cases

SAS, SSS

Law of Cosines

Triangle ABC with side lengths a, b, and c

• e.g. a²=b²+c²-2bc ⋅ cosA