RIGHT TRIANGLES Geometric Mean n When a b
RIGHT TRIANGLES
Geometric Mean n When a, b, and x are positive numbers and a/x = x/b, x is called the geometric mean between a and b.
Example n Find the geometric mean between 3 and 6.
Theorem n The altitude to the hypotenuse of a right triangle forms right triangles that are similar to each other and to the original triangle.
Theorem n In a right triangle, the length of the altitude to the hypotenuse is the geometric mean between the lengths of the two segments on the hypotenuse.
Theorem n In a right triangle with an altitude to the hypotenuse, each leg is the geometric mean between the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
Pythagorean Theorem
Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. n c 2 = a 2 + b 2 n
Find the missing length 1. a = 6, b = 8 2. c = 15, a = 9 3. b = 2, a = 2
Solve n The base of an isosceles triangle is 2 x cm long. The altitude to the base is 3 x cm long. Find the length of one other side of the triangle.
Solve n Find the altitude of an equilateral triangle with side length ten.
Solve n Find the perimeter of a rectangle that has diagonal length 8 and side length 5.
Converse of Pythagorean Theorem
Converse of the Pythagorean Theorem n If a triangle has side lengths a, b, and c and a 2 + b 2 = c 2, then the triangle is a right triangle with right angle opposite the side length c.
Question 1 n Is a triangle with sides 8, 15 and 17 a right triangle?
Question 2 n Is a triangle with sides 5, 6 and 8 a right triangle?
Theorem If a < b < c are lengths of the sides of a triangle and 1. a 2 + b 2 < c 2, then the triangle is an obtuse triangle. 2. a 2 + b 2 > c 2, then the triangle is an acute triangle. n
Question 1 n Is a triangle with sides of lengths 2, 3 and 4 acute, right or obtuse?
Question 2 Which of these triples are sides of a right triangle? a. (2, 3, 4) b. (6, 8, 10) c. (1, 1, 2) d. (0. 1, 0. 4, 0. 5) n
Pythagorean Triples
Pythagorean Triples n A Pythagorean triple is any three whole numbers a, b, and c that satisfy the equation a 2 + b 2 = c 2. These numbers are the sides of a right triangle. Some commonly used Pythagorean triples are (3, 4, 5), (8, 15, 17) and (7, 24, 25). Any multiple of a Pythagorean triple is also a Pythagorean triple. (6, 8, 1), (9, 12, 15)….
Pythagorean Triples n If n is a positive integer, then (2 n + 1, 2 n 2 + 2 n + 1) is a Pythagorean triple.
Special Right Triangles
45 -45 -90 Triangle n The length of the hypotenuse of a 45 -45 -90 triangle is square root of two times the length of a leg.
Question 1 n Find the length of the diagonal of a square with sides 8 cm long.
Question 2 n If a perimeter of a square is 64 cm, how long is its diagonal?
30 -60 -90 Triangle n In a 30 -60 -90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg and the length of the longer leg is square root of three times the length of the shorter leg.
Question 1 n Find the length of an altitude of an equilateral triangle with perimeter 24.
Question 2 n A ladder leaning against a wall makes a 60 angle with the ground. The base of the ladder is 3 m from the building. How high above the ground is the top of the ladder?
n Check if all pairs have a sum greater than the remaining side
- Slides: 30