Lesson - Right Triangles and the Pythagorean Theorem
 The Pythagorean Theorem can be usedto find the length of the missing side ofa right triangle if you know the lengthof the other two sides.512X The formula for the Pythagorean Theorem is:                        a2 + b2 = c2In words, we would say:"The sum of the squares of the legs is equalto the square of the hypotenuse."512X In order to understand all of this,we need to know what "a," "b," and "c" mean and what "LEGS" and the "HYPOTENUSE" mean. All triangles have three sides.Right triangles have special names for their sides.The two sides that meet at the right angle are called"LEGS" and their lengths are represented by a and b.The side opposite the right angle is called the"HYPOTENUSE" and its length is represented by c.LEGHYPLEGOTENUSaEbc It doesn't matter which way the triangle is facing.a and b are always the legs of the triangle andc is always the hypotenuse.cab zWhich side is the hypotenuse?(click on the correct answer)xxyyz zIf you picked y, you were correct!!!xxyyz So, now you know how to find the legs (a and b)and the hypotenuse (c) of a right triangle.Let's review the theorem again:The sum of the squares of the legs (a and b)                 a2    +    b2    =    c2is equal to the square of the hypotenuse (c). So how do we use thisPythagorean Theorem to findthe missing side of a right triangle?Well, we've covered the first step already.First we have to identify the legs and thehypotenuse.The legs have lengths 3 and 4 and we don'tknow how long the hypotenuse is.x43 Once we've identified the legs and hypotenuse,we substitute these for the letters in the formula.a = 3b = 4c = xa2 + b2 = c2(3)2 + (4)2 = (x)2 x43 Then we simplify the expression usingthe order of operations (PEMDAS).First we'll handle the exponents,then we'll add.(3)2 + (4)2 = (x)2 (3)(3) + (4)(4) = x2 9 + 16 = x225 = x2THE ANSWER IS NOT 25!!! Now that the equation is written as simply aspossible we can solve it. We solve equations byisolating the variable(getting the variable alone on one side of theequation). We use opposite operations to do this.In order to isolate the variable in this equation,we have to get rid of the exponent. The oppositeof the exponent of 2 (squared) is the square root.To find the square root we ask:"Which number, multiplied by itself producesthe number under the square root symbol?"√25 = √x2              5 = xSince 5 times 5 equals 25, the square root of 25 is 5. Let's review the steps:1. Identify the legs    and hypotenuse.2. Substitute their lengths    into the formula.3. Simplify the formula.4. Solve for the variable.5. Write the answer: The hypotenuse is 5 units long.a = 3b = 4c = xa2 + b2 = c2(3)2 + (4)2 = (x)2 (3)(3) + (4)(4) = x2 9 + 16 = x225 = x2√25 = √x2              5 = xx43 In the previous example, the hypotenuse wasmissing. In some cases, however, one of thelegs will be missing. Let's see how that works:When one of the legs is missing, it doesn'tmatter which letter you use. Both of these arecorrect:a = h                   or              a = 40b = 40                                  b = hc =41                                    c = 41h4041 a = hb = 40c = 41a2 + b2 = c2(h)2 + (40)2 = (41)2h2 + (40)(40) = (41)(41)h2 + 1600 = 1681     - 1600    -1600h2 = 81√h2 = √81h = 9h4041Here is where it is different:To solve, first you subtract,then you take the square root.The missing leg is 9 units long. Now you try one:Which way should you substitute the numbers?                          (Pick one)(8)2 + (6)2 = (n)268n(6)2 + (n)2 = (8)2 Which is the correct simplification of the equation?                                (PICK ONE)If you picked: (8)2 + (6)2 = (n)2               YOU'RE CORRECT!!!You could also write: (6)2 + (8)2 = (n)2  28 = n2  100 = n2  196 = n2 How long is the hypotenuse (c)?               PICK ONE1005010100 = n2is CORRECT!!! √10 = √n210 = nThe hypontenuse is 10 units long. Let's try one more:Which way should you substitute the numbers?                          (Pick one)(13)2 + (5)2 = (m)2135m(m)2 + (5)2 = (13)2 Which is the correct simplification of the equation?                                (PICK ONE)If you picked: (m)2 + (5)2 = (13)2               YOU'RE CORRECT!!!You could also write: (5)2 + (m)2 = (13)2  m2 + 25 = 169  m2 + 10 = 26 What is the next step and the result?                   PICK ONE  Take the square root of both sides.      m + 5 = 13  Subtract 25.      m2 = 144m2 + 25 = 169is CORRECT!!! How long is the missing leg?            PICK ONE  You must subtract 25.      m2 = 144  72 units  12 units  144 units √m2 = √144m = 12The leg is 12 units long.    THE ENDGOOD LUCK!
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