Pythagorean Theorem Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

Right Triangles A right triangle is a triangle that has one right angle in it. The hypotenuse of a right triangle is the side that is opposite from the right angle. It is always the longest side in the triangle. The other two smaller sides are called legs.

This side is the hypotenuse since it is opposite from the right angle. Right Triangles This side is the hypotenuse since it is opposite from the right angle. Leg The other two sides are then called legs. Leg

Definition The Pythagorean Theorem is a mathematical theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs. In a right triangle like the one below, c a2 + b2 = c2 a b

Let’s find the length of the missing side. Example Let’s find the length of the missing side. The side that is not given is the hypotenuse since it is opposite from the right angle. In other words this would be our side “c”. x 3 4 Substitute values in for a, b, and c into the Pythagorean Theorem. a2 + b2 = c2 (3)2 + (4)2 = x2

Let’s find the length of the missing side. Example Let’s find the length of the missing side. You must solve the equation algebraically to find the value for x. x 3 4 x is almost by itself. We must get rid of the square. What is the opposite of squaring a number? (3)2 + (4)2 = x2 9 + 16 = x2 25 = x2 A square root is the opposite. So you must take the square root of both sides to get rid of the square. 5 = x

Let’s find the length of the missing side. Example Let’s find the length of the missing side. You can check your answer to see if it is correct by substituting your answer back in for x. 5 3 4 (3)2 + (4)2 = x2 (3)2 + (4)2 = (5)2 9 + 16 = 25 25 = 25 ✓

Example #2 Let’s take a look at an example where the value for the hypotenuse is given. This time the hypotenuse is 13 and we are not given the value for one of the legs. What do you think the correct values for a, b, and c should be? Substitute those in. 13 x a2 + b2 = c2 5 (5)2 + x2 = (13)2 Simplify the equation as much as you can before you start solving for x.

Example #2 Let’s take a look at an example where the value for the hypotenuse is given. (5)2 + x2 = (13)2 25 + x2 = 169 Now solve for x. - 25 - 25 13 x x2 = 144 5 x = 12 Check your work to see if 12 is the correct answer!!

Shortcuts There are two formulas you could use as shortcuts. The first case is when you are given the lengths of the two legs and you must find the length of the hypotenuse… Use : The second case is when you are given the hypotenuse and a leg and you must find the length of the second leg… Use : OR

Example #4 Let’s use the formulas to find the missing sides for the next two examples. 7 Are we trying to find a leg or a hypotenuse? We are finding a hypotenuse since x is opposite of the right angle. x 24 What formula should we use?

Example #4 Let’s use the formulas to find the missing sides for the next two examples. 7 Substitute the values in for a, b, and c. If you substitute the values in correctly you should get. x 24

Example #5 Find the length of x, using one of the formulas. 24 26 In this situation you must make sure that you use 26 for c. Remember c represents the hypotenuse (which is always the largest side). Which formula should you use?

Squares and Diagonals The diagonal of a square is a line segment that connects a square’s opposite vertices (corners). When asked to find the length of a diagonal of a square, you must use the Pythagorean Theorem.

Squares and Diagonals Find the length of the diagonal of the following square. Round to the nearest hundredth. 16 First you should note that all sides of a square are all the same value. Also, all of its angles are right angles. 16 16 Draw in one of the diagonals of the square. 16

Squares and Diagonals Find the length of the diagonal of the following square. Round to the nearest hundredth. 16 You should notice that a diagonal splits a square into two right triangles. 16 16 16

Squares and Diagonals Find the length of the diagonal of the following square. Round to the nearest hundredth. You should notice that a diagonal splits a square into two right triangles. 16 16

Squares and Diagonals Find the length of the diagonal of the following square. Round to the nearest hundredth. Let’s look at only one of the triangles. Find the missing side using the Pythagorean Theorem. x 16 16

Pythagorean Triples Pythagorean triples are three whole numbers that form a right triangle. In other words, if you square both the smaller numbers and add them together, that number should be equal to the largest number squared. Ex: Is a triangle with the sides 5, 3, and 4 a right triangle? Which two sides are the smallest? 3 and 4

Pythagorean Triples Pythagorean triples are three whole numbers that form a right triangle. In other words, if you square both the smaller numbers and add them together, that number should be equal to the largest number squared. Ex: Is a triangle with the sides 5, 3, and 4 a right triangle? Square the two legs and add them together. If these values form a right triangle, that number should equal the hypotenuse squared. Legs: 3 and 4 Hypotenuse: 5

Pythagorean Triples Pythagorean triples are three whole numbers that form a right triangle. In other words, if you square both the smaller numbers and add them together, that number should be equal to the largest number squared. Ex: Is a triangle with the sides 5, 3, and 4 a right triangle? Legs: 3 and 4 Hypotenuse: 5 Legs: 32 + 42 = 25 They are both equal to 25, so it is a right triangle. Hypotenuse: 52 = 25

Pythagorean Triples Example 2: Is 10, 14, 9 a right triangle? What are the values of the legs and hypotenuse? Legs: 9 and 10 Hypotenuse: 14 92 + 102 = 181 142 = 196 They are not equal, so the three numbers do not form a right triangle.

Follow-Up Questions Answer the following questions on loose leaf and hand them in to your teacher.

Find the values for h or x. Round to the nearest hundredths. Follow-Up Questions Find the values for h or x. Round to the nearest hundredths. 1) 2) 3) 4)

Follow-Up Questions How far up a wall will an 11m ladder reach, if the foot of the ladder must be 4m from the base of the wall? What is the length of a diagonal of a square whose sides are 16 in long. Is a triangle with side lengths 25, 20, 15 a right triangle? Is a triangle with side lengths 7, 7, 10 a right triangle?