Geometry Right Triangle Vocabulary

Trigonometric Ratio

A ratio of the lengths of two sides in a right triangle. Three common trigonometric rations are sine, cosine, and tangent.

Pythagorean triple

A set of three positive integers a, b, and c that satisfy the equation c² = a² + b². Common Pythagorean triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 *Their multiples as well

Cosine

A trigonometric ratio, abbreviated as cos. For a right triangle ABC, the cosine of the acute angle A is cos A = length of leg adjacent to ∠A / length of leg hypotenuse to ∠A = AC/AB.

Sine

A trigonometric ratio, abbreviated as sin. For a right triangle ABC, the sine of the acute angle A is sin A = length of leg opposite ∠A / length of hypotenuse = BC/AB.

Tangent

A trigonometric ratio, abbreviated as tan. For a right triangle ABC, the tangent of the acute angle A is tan A = length of leg opposite ∠A / length of leg adjacent to ∠A = BC/AC.

Inverse Cosine

An inverse trigonometric ratio, abbreviated as cos. For acute angle A, if cos A = z, then cos^-1 z = m∠A.

Inverse Sine

An inverse trigonometric ratio, abbreviated as sin^-1. For acute angle A, if sin A = y, then sin^-1 y = m∠A

Inverse Tangent

An inverse trigonometric ratio, abbreviated as tan^-1. For acute angle A, if tan A = x, then tan^-1 x = m∠A

Theorem 7.5

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

Theorem 7.2 Converse of the Pythagorean Theorem

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If c² = a² + b², then ΔABC is a right triangle.

Theorem 7.4

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. If c² > a² + b², then triangle ABC is obtuse.

Theorem 7.3

If the square of the length of the longest sides of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. If c² < a² + b², the the triangle is acute.

Theorem 7.9 30º-60º-90º Triangle Theorem

In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. hypotenuse = 2 ∙ shorter leg longer leg = shorter leg ∙ √3

Theorem 7.8 45º-45º-90º Triangle Theorem

In a 45º-45º-90º triangle, the hypotenuse is √2 times as long as each leg. hypotenuse = leg ∙ √2

Theorem 7.7 Geometric Mean (Leg) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

Theorem 7.6 Geometric Mean (Altitude) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

Theorem 7.1 Pythagorean Theorem

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

Solve a Right Triangle

To find the measures of all the sides and angles of a right triangle. You can solve a right triangle if you know either of the following: - Two sides lengths - One side length and the measure of one acute angle

Angle of Depression

When you look down at an object, the angle that your line of sight makes with a line drawn horizontally.

Angle of Elevation

When you look up at an object, the angle that your line of sight makes with a line drawn horizontally.