Apex Geometry Unit 4: Right Triangles

Common Pythagorean Triples and Some of their Multiples

3, 4, 5 6, 8, 10 9, 12, 15 3x, 4x, 5x 5, 12, 13 10, 24, 26 15, 36, 29 8, 15, 17 16, 30, 34 24, 45, 51 8x, 15x, 17x 7, 24, 25 14, 48, 50 21, 72, 75 7x, 24x, 25x

Trigonometric Ratio

A ratio of the lengths of two sides in a right triangle to the angle.

Standard Position

A right triangle whose hypotenuse is a radius of the circle of radius 1 with center at the origin of a coordinate plane with one leg on the x-axis and the other leg perpendicular to the x-axis.

Pythagorean Triple

A set of three positive integers that satisfy the equation a²+b²=c².

Cosine Ratio (CAH)

A trigonometric ratio for acute angles that involves the length of the adjacent leg of the angle in a right triangle and its hypotenuse. Length of leg adjacent to the angle/length of the hypotenuse.

Sine Ratio (SOH)

A trigonometric ratio for acute angles that involves the length of the opposite leg of the angle in a right triangle and its hypotenuse. Length of leg opposite the angle /length of the hypotenuse.

Tangent Ratio (TOA)

A trigonometric ratio for acute angles that involves the lengths of the legs of a right triangle. Length of leg opposite the angle /length of the leg adjacent the angle

Solving a Right Triangle

Finding all side lengths and angle measures of a triangle.

Pythagorean Inequalities Theorem

For any ∆ABC, where c is the length of the longest side, the following statements are true: If a²+b²>c², then ∆ABC is acute. If a²+b²<c², then ∆ABC is obtuse.

Right Triangle Similarity Theorem

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.

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.

Theorem 9.5: 30°-60°-90° Triangle Theorem

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

Theorem 9.4: 45°-45°-90° Triangle Theorem

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

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. a²+b²=c², where c is always the hypotenuse.

Area of a Triangle

The area of any triangle is given by one-half the product of the lengths of two sides times the sine of their include angle.

Inverse Trigonometric Ratios

You can use sin⁻¹, cos⁻¹ and tan⁻¹ to find the measure of an angle in a right triangle given its sides.