Honors Geometry chapter 8

Pythagorean inequality theorem (obtuse) \[def\]

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.

Geometric mean

________ (leg) theorem [def]- the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments.

Pythagorean theorem

________ (def)- the sums of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse.

Pythagorean inequality theorem (acute) \[def\]

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 2 sides, then the triangle is acute.

geometric mean

x² = ab; x = √ab

geometric mean (altitude) theorem [def]

the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments

geometric mean (altitude) theorem [equation]

segment 1 / altitude = altitude / segment 2

geometric mean (leg) theorem [def]

the altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments

geometric mean (leg) theorem [equation]

hypotenuse / leg = leg / corresponding segment

pythagorean theorem (def)

the sums of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse

pythagorean theorem (equation)

a² + b² = c²

pythagorean triple

three integers that follow the pattern of the pythagorean theorem

converse of the pythagorean theorem [def]

if the longest side squared is equal to the sum of the other sides squared in a triangle, then it is a right triangle

converse of the pythagorean theorem [equation]

c² = a² + b² = right triangle

pythagorean inequality theorem (acute) [def]

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 2 sides, then the triangle is acute

pythagorean inequality theorem (acute) [equation]

c² < a² + b² = acute

pythagorean inequality theorem (obtuse) [def]

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

pythagorean inequality theorem (obtuse) [equation]

c² > a² + b² = obtuse

angle measures for an isosceles right triangle

45-45-90

angle measures for a non-isosceles right triangle

30-60-90

side measures for a 45-45-90 triangle

x (leg) - x (leg) - x√2 (hypotenuse)

side measures for a 30-60-90 triangle

x (shorter leg) - x√3 (longer leg) - 2x (hypotenuse)