Math 2 Chapter 7, Lessons 1 & 2

You plan to build a path along one diagonal of a 100 ft-by-100 ft square garden. To the nearest foot, how long will the path be?

100 x square root 2 141 ft

The size of a computer monitor is the length of its diagonal. You want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor? (round to the nearest tenth of an inch)

11 squared + x squared = 19 squared 121 + x squared = 361 -121 -121 x squared = 240 take square root of both sides x = 15.5 inches

Converse of the Pythagorean Theorem

If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triagnle is a right triangle.

What are the qualifications for a Pythagorean triple?

1. All whole numbers 2. must satisfy Pythagorean Theorem Example: 3, 4, 5

The legs of a right triangle have lengths of 10 and 24. What is the length of the hypotenuse? Is this a Pythagorean triple?

10 squared + 24 squared = c squared 100 + 576 = c squared take square root of 676 c = 26 Yes, they are all whole numbers and they satisfy the Pythagorean Theorem.

A triangle has side lengths of 16, 48, and 50. Is the triangle a right triangle?

16 squared + 48 squared = 50 squared 256 + 2304 = 2500 2560 = 2500 No, the answers have to be equal.

The length of the hypotenuse of a 45-45-90 triangle is 16. What is the length of one leg?

16/square root 2 = leg x square root 2/square root 16/square root 2 x square root 2/square root 2 = 16 square root 2/2 8 square root 2

Is a triangle with side lengths 7, 8, and 9 acute, obtuse, or right?

49 + 64 = 81 113 > 81 acute

What is the length of the hypotenuse of a 45-45-90 tiangle with leg length of 5 square root 3?

5 square root 3 x square root 2 5 square root 6

The hypotenuse of a right triangle has a length of 12. One leg has a length of 6. What is the length of the other leg? (simplest radical form)

6 squared + x squared = 12 squared 36 + x squared = 144 -36 -36 x squared = 108 take square root of 108 x = 6 square root 3

Pythagorean Theorem

Pythagoream Theorem: if the triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse leg 1 + leg 2 = hypotenuse (a squared + b squared = c squared)

Acute or Obtuse? c squared < a squared + b squared

acute

45-45-90 Triangle Theorem

both legs are congruent and the length of the hypotenuse is square root 2 times the length of a leg hypotenuse = square root 2 x leg

What segment is always the biggest on a right triangle?

hypotenuse

Once you know which length represents the hypotenuse, does it matter which length you substitute for a and which length you substitute for b?

no

Acute or Obtuse? c squared > a squared + b squared

obtuse

For real life problems, do you express your answer in radical form or in irrational, rounded numbers?

rounded numbers

Example: The hypotenuse of a 45-45-90 triangle is 2 square root 3. What is the length of one leg?

square root 6

30-60-90 Triangle Theorem

the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is square root 3 times the length of the shorter leg hypotenuse = 2 x shorter leg longer leg = square root 3 x shorter leg