To get the patterns for a 45-45-90 triangle, it's better to use a square that has equal side of 1. From there we can split the square diagonally and get two 45-45-90 triangles. From there we can use the Pythagorean theorem to get the hypotenuse of the triangle and from there we can see the pattern starting to form. Now we use "n" to represent the ratio of each side and as a variable to represent any number that can take it's place. These next few pictures will show how we can derive the pattern of the 45-45-90 triangle.
THESE SET OF PICTURES ARE FOR A 45-45-90 TRIANGLE ONLY:
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| Here we have a perfect square with each side being 1 and each corner being 90 degrees. To get the 45-45-90 triangle we are going to do one step to get the triangle and the Pythagorean theorem to get the missing side. |
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| To get our triangles we just split the square diagonally but we are just going to use the triangle highlighted in green. We already have two sides, the horizontal and the vertical side, and each side is 1. The triangle is still incomplete because we need the hypotenuse and to find it we'll use the Pythagorean theorem to find it. |
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| It doesn't matter which side is a or b because both sides are the same. We plug it in the the Pythagorean theorem and we have 1^2 + 1^2 = c^2. We square the 1's and add them and we get 2 on that side. Now it's 2 = c^2, we square root each side making 2 into radical 2 and c^2 to c. So radical 2 will be our hypotenuse for the 45-45-90 triangle. We nearly done but we need to plug in "n" to each side, so the vertical and the horizontal sides will be "n" and "n-radical-2" for the hypotenuse side. |
THESE SET OF PICTURES ARE FOR A 30-60-90 TRIANGLE.
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| Here we have an equilateral triangle with each side length of 1 and each angle being 60 degrees. To get a 30-60-90 triangle we split the triangle in half and get two of them but in this case we are only going to use the one highlighted green. The hypotenuse is already there so it's length is 1, the horizontal is also there but it's not 1 because we split it in half so it's actually 1/2. The third side will be found by using the Pythagorean theorem. |
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| In this case, side "a" will be the horizontal side and side "b" will be the vertical side. When we plug it into the Pythagorean theorem, 1/2 will be squared and it will be 1/4 and 1^2 will just be 1. So we then subtract 1/4 to both sides and end up with b^2 = 3/4 but we need to square root both side to make b^2 to just b. When we square root 3/4 it applies to the top and the bottom, which means that 3 will be radical-3 and 4 will be 2 since the square root of 4 is 2. Our final answer for side"b" will be radical-3/4. |
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| This is how a 30-60-90 triangle is suppose to look, the hypotenuse side being 2n, side "a" being n, and side "b" being n-radical-3. Well when I left off in the other picture it was suppose to be radical-3 for side "b", 1/2 for side "a", and 1 for the hypotenuse. To get it to be the derived pattern we just multiply 2 to each side, so the hypotenuse will be 2, side "a" 1 because the 2's cancel each other, and radical-3 for side "b" because the 2's also cancel each other. The "n's" are put in to show that any number can take it's place. |
INQUIRY ACTIVITY REFLECTION
1. Something I never noticed before about special right triangles is how they are derived form other shapes like the square and the equilateral triangle.
2. Being able to derive these patterns myself aids in my learning because it shows that I know how to derive them and how they came to be a 45-45-90 and 30-60-90 triangle.
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