1. Find the area and perimeter of the rectangle.

2. Simplify each radical expression:

3. Algebra 1 Glencoe McGraw-Hill JoAnn Evans Math 8H 10-4 The Pythagorean Theorem 10-5 The Distance Formula

4. The Pythagorean Theorem is a statement that describes the relationship between the three sides of a right triangle. (hypotenuse) (leg) This is Pythagoras, a Greek mathematician who lived from about 585-500 B.C. Although the theorem is named after him, there are indications that the theorem was in use in northern Africa before Pythagoras wrote of it. a b c a 2 + b 2 = c 2 For any right triangle, the sum of the squares of the lengths of the legs, a and b, equals the square of the length of the hypotenuse, c.

5. A right triangle has one leg that is 2 inches longer than the other leg. The hypotenuse is 10 inches. Find the unknown lengths. x 10 a 2 + b 2 = c 2 x + 2 Length is positive, so one length is 6” and the other length is 8”.

6. Determine whether the following side measures form a right triangle. Justify your answer. 7, 24, 25 If a triangle is a right triangle, then the three sides will satisfy the equation How do you know which two measurements are the sides and which is the hypotenuse? The hypotenuse (the side opposite the right angle formed by the legs) will always be the longest measurement.

7. Determine whether the following side measures form a right triangle. Justify your answer. 16, 28, 32 5, 7,

8. The distance formula is a tool used extensively in engineering, navigation, the natural sciences, and in any environment that uses coordinate geometry. This useful formula is derived from the Pythagorean Theorem. Pythagorean Theorem Distance Formula

9. 4 3 c Find the distance between the points (2, 1) and (6, 4). 1.Plot the two points. 2.Count the horizontal distance between the two points. 3.Count the vertical distance between the two points. 4.Use the Pythagorean Theorem to find the unknown distance. (2,1) (6,4) Because distance is positive, only the positive root will be considered. 5 Take the square root of both sides to solve for c.

10. 4 3 (2,1) (6,4) 5 Let d = the distance between points (2, 1) and (6, 4). The length of leg a was found by taking the difference between the x-coordinates of (6,4) and (2,1). The length of leg b was found by taking the difference between the y-coordinates of (6,4) and (2,1).

11. Using the distance formula, find the distance between the points (4, 5) and (-2, 6). Let point (4, 5) be (x 2, y 2 ) Let point (-2, 6) be (x 1, y 1 ) Stack the two points like you learned to do when calculating slope. It doesn’t matter which point you choose to put on top. Either way, the distance formula give the same answer.

12. Using the distance formula, find the distance between the points (-2, 10) and (5, -3). Let point (-2, 10) be (x 2, y 2 ) Let point (5, -3) be (x 1, y 1 )

13. Using the distance formula, find the distance between the points (-14, -4) and (1, 8). Let point (-14, -4) be (x 2, y 2 ) Let point (1, 8) be (x 1, y 1 )

14. Using the distance formula, find the distance between the points and.

15. Find the possible values of a if the distance between the points at (4, -1) and (a, 5) is 10 units. Let point (4, -1) be (x 2, y 2 ) Let point (a, 5) be (x 1, y 1 ) Substitute. Simplify. Evaluate squares. Simplify.

16. Square each side. Subtract 100 from each side. Factor. The value of a is 12 or -4.

17. (4,-1) (12,5)(-4,5) The distance formula calculations show that there are two points with a y value of 5 that are 10 units away from the point (4, -1). (a,5) 10

18. Find the possible values of a if the distance between the points at (-9, 7) and (a, 5) is. Let point (-9, 7) be (x 2, y 2 ) Let point (a, 5) be (x 1, y 1 ) Substitute. Simplify. Evaluate squares. Simplify.

19. Square each side. Subtract 29 from each side. Factor. The value of a is -14 or -4.