April 09, 2012 "Perfect Squares" On the grid below, you can see that each square has sides with integer length. The area of a square is the square of the length of a side. A = s2 The square of an integer is a perfect square! April 09, 2012 Square Roots and Irrational Numbers The inverse of squaring a number is finding a square root. The square-root radical, , indicates the nonnegative square root of a number. The number underneath the square root (radical) sign is called the radicand. Ex: 16 = 121 = 49 = 144 = The square of an integer results in a perfect square. Since squaring a number is multiplying it by itself, there are two integer values that will result in the same perfect square: the positive integer and its opposite. 2 Ex: If x = (perfect square), solutions would be x and (-x)!! Ex: a2 = 25 5 and -5 would make this equation true! April 09, 2012 If an integer is NOT a perfect square, its square root is irrational!!! Remember that an irrational number has a decimal form that is non- terminating, non-repeating, and thus cannot be written as a fraction. For an integer that is not a perfect square, you can estimate its square root on a number line. Ex: 8 Since 22 = 4, and 32 = 9, 8 would fall between integers 2 and 3 on a number line. -2 -1 0 1 2 3 4 5 April 09, 2012 Approximate where √40 falls on the number line: Approximate where √192 falls on the number line: April 09, 2012 Topic Extension: Simplest Radical Form Simplify the radicand so that it has no more perfect-square factors. April 09, 2012 Right Triangles and The Pythagorean Theorem On your graph paper, draw two right triangles: 1) 3 units by 4 units, then connect the ends. 2) 6 units by 8 units, then connect the ends. Next, use the second sheet of graph paper to measure the length of the side opposite the right angle. a c b c a b Shade a square with dimensions equal to the side- lengths of the two shortest sides of the triangles. April 09, 2012 In a right triangle, the two shortest sides are called legs. The longest side (opposite the right angle), is the hypotenuse. Pythagorean Theorem In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. (leg)2 + (leg)2 = (hypotenuse)2 a2 + b2 = c2 c a Hypotenuse Legs b April 09, 2012 Draw some right triangles on your graph paper and test the Pythagorean Theorem to find the length of the hypotenuse. Round to the nearest tenth. Ex: In a right triangle, the length of the hypotenuse is 15 m and the length of a leg is 8 m. Find the length of the other leg to the nearest tenth. April 09, 2012 The Converse of the Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the equation leg2 + leg2 = hypotenuse2 or (a2 + b2 = c2 ) to check whether a triangle is a rt. triangle. If the equation is true, the triangle is a right triangle. Ex: 7 in., 8 in., 113 in. Ex: 5 mm, 6 mm, 10 mm April 09, 2012 Proof of Pythagorean Theorem and Practice hypotenuse leg leg April 09, 2012 Find the height of the diving board, rounding to the nearest tenth!.