Answers:
- 30-60-90 triangle = 4, 8, 4*sqrt(3)
- 45-45-90 triangle = 6, 6, 6*sqrt(2)
- Right triangle, but not special right triangle = 6, 8, 10
- Triangle but not a right triangle = 7, 8, 10
- Not a triangle = 7, 12, 19
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Explanation:
The template for a 30-60-90 triangle is that the short leg x leads to hypotenuse 2x, and longer leg x*sqrt(3). If x = 4, then we get the sides 4, 8, 4*sqrt(3) which are the short leg, hypotenuse, and long leg in that order.
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The template for a 45-45-90 triangle is that we have x as the leg length and x*sqrt(2) as the hypotenuse. If x = 6, then we end up with the sides 6, 6, 6*sqrt(2)
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The triangle with sides 6, 8, 10 is a right triangle because a = 6, b = 8, c = 10 makes a^2+b^2 = c^2 true. I'm using the converse of the pythagorean theorem. We don't have a special right triangle because these sides do not fit the templates mentioned in the previous two sections above.
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The triangle with sides a = 7, b = 8, c = 10 on the other hand will not make a^2+b^2 = c^2 true. Note that a^2+b^2 = 7^2+8^2 = 113 which is not the same as 10^2 = 100. However, a triangle is possible because adding any two sides leads to a sum larger than the third side (triangle inequality theorem).
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We cannot form a triangle with sides 7, 12, 19 because the sides 7 and 12 add to 7+12 = 19 but that's not longer than the third side 19.
The triangle inequality theorem says that if you want to form a triangle with sides a,b,c then the following three inequalities must all be true
a+b > c
a+c > b
b+c > a
