Geometric Mean Pythagorean Theorem Special Right Triangles

Geometric Mean When the means of a proportion are the same number, that number is called the geometric mean of extremes.

Geometric Mean The geometric mean between two numbers is the positive square root of their product.

Examples Find the geometric mean between the pair of numbers. 5 and 45

Examples Find the geometric mean between the pair of numbers. 5 and 45

Examples Find the geometric mean between the pair of numbers. 12 and 15

Examples Find the geometric mean between the pair of numbers. 12 and 15 x2 = 12 * 15 = 180 x = √180 = 6√5

Geometric Means in Triangles In a right triangle, an altitude drawn from the vertex of the right triangle to the hypotenuse forms two additional right triangles. These three right triangles share a special relationship.

Geometric Means in Triangles 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.

Geometric Mean (Altitude) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this altitude is the geometric mean between the lengths of these two segments.

Geometric Mean (Leg) Theorem The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a leg of this triangle is the geometric mean between the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Geometric Mean (Leg) Theorem

Examples Find x, y, and z.

Examples Find x, y, and z. z = √8*25 = √200 z = 10√2

Examples Find x, y, and z.

Examples Find x, y, and z. 12 = √(9*x) 144 = 9x 16 = x y = √(16*25)

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

Pythagorean Triple A Pythagorean triple is a set of three nonzero whole numbers a, b, and c, such that a2 + b2 = c2.

Examples Use a Pythagorean triple to find x. Explain your reasoning.

Examples Use a Pythagorean triple to find x. Explain your reasoning. 20 = 5*4 48 = 12*4 x = 13*4 = 52

Pythagorean Inequality Theorem If the square of the length of the hypotenuse is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.

Pythagorean Inequality Theorem If the square of the length of the hypotenuse is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle.

45-45-90 Triangle Theorem In a 45-45-90 triangle, the legs l are congruent and the length of the hypotenuse h is √2 times the length of a leg.

Examples Find x.

Examples Find x. x = 5√2

Examples Find x.

Examples Find x. 18 = x√2 x = 18/√2 x = 9√2

Examples Find x.

Examples Find x. x = 7√2 *√2 x = 7*2 x = 14

30-60-90 Triangle Theorem In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg and the longer leg is √3 times the length of the shorter leg.

Examples Find x and y.

Examples Find x and y. 15 = x√3 x = 15/√3 x = 5√3 y = 2x

Examples Find x and y.

Examples Find x and y. x = 12/√3 x = 4√3 y = 2*x y = 2 * 4√3 y = 8√3

Examples Find x and y. x = .5 * 10 x = 5 y = x * √3 y = 5√3