Lesson - Right Triangles and the Pythagorean Theorem

The Pythagorean Theorem can be used to find the length of the missing side of a right triangle if you know the length of the other two sides. 5 12 X The formula for the Pythagorean Theorem is:a ^{2} + b^{2} = c^{2}In words, we would say:"The sum of the squares of the legs is equalto the square of the hypotenuse."5 12 X 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. L E G H Y P LEG O T E N U S a E b c 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. c a b z Which side is the hypotenuse? (click on the correct answer) x x y y z z If you picked , you were correct!!!yx x y y z 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) a ^{2} + b^{2} = c^{2}is equal to the square of the hypotenuse (c). So how do we use this Pythagorean Theorem to find the missing side of a right triangle? Well, we've covered the first step already. First we have to identify the legs and the hypotenuse. The legs have lengths 3 and 4 and we don't know how long the hypotenuse is. x 4 3 Once we've identified the legs and hypotenuse, we substitute these for the letters in the formula. a = 3b = 4c = x a ^{2} + b^{2} = c^{2}(3) ^{2} + (4)^{2} = (x)^{2} x 4 3 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) = x ^{2} 9 + 16 = x ^{2}25 = x ^{2}THE ANSWER IS 25!!!NOTNow that the equation is written as simply aspossible we can solve it. We solve equations by (getting the variable alone on one side of theequation). We use opposite operations to do this.isolating the variableIn 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 = √x ^{2} 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 = xa ^{2} + b^{2} = c^{2}(3)^{2} + (4)^{2} = (x)^{2} (3)(3) + (4)(4) = x^{2} 9 + 16 = x^{2}25 = x^{2}√25 = √x^{2} 5 = xx 4 3 In the previous example, the hypotenuse was missing. In some cases, however, one of the legs 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 = 41 h 40 41 a = hb = 40c = 41a ^{2} + b^{2} = c^{2}(h)^{2} + (40)^{2} = (41)^{2}h^{2} + (40)(40) = (41)(41)h^{2} + 1600 = 1681 - 1600 -1600h^{2} = 81√h^{2} = √81h = 9 h 40 41 Here 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)^{2}6 8 n (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 = n ^{2} 100 = n ^{2} 196 = n ^{2}How long is the hypotenuse (c)? PICK ONE100 50 10 100 = n ^{2}is CORRECT!!! √10 = √n2 10 = n The hypontenuse is 10 units long. Let's try one more: Which way should you substitute the numbers? (Pick one)(13) ^{2} + (5)^{2} = (m)^{2}13 5 m (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} m ^{2} + 25 = 169 m ^{2} + 10 = 26What is the next step and the result? PICK ONE Take the square root of both sides. m + 5 = 13 Subtract 25. m ^{2} = 144m ^{2} + 25 = 169is CORRECT!!!How long is the missing leg? PICK ONE You must subtract 25. m ^{2} = 144 72 units 12 units 144 units √m ^{2} = √144m = 12 The leg is 12 units long. THE ENDGOOD LUCK! |

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