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8) Which of the following inequalities is true about the lengths a and b of the sides of the triangle above? (fig. not drawn to scale)= shows a right triangle, a and b are legs, and 20 is the hypotenuse. The answer is 400 less than or equal to (a + b)^2 .
Right triangles are delicious.
Well, we know a^2 + b^2 = 400, right? Pythagoras told us that.
If you foil out (a + b)^2, you get a^2 + b^2 + 2ab. Since we know a and b are positive (they have to be — they’re lengths of triangle legs), we know that the 2ab makes that value bigger.
So (a + b)^2 is bigger than a^2 + b^2 = 400. Wacky, right?
You could also, if you want to plug in, think of a Pythagorean triple to use. If the hypotenuse is 20, you can use 12-16-20, the big brother of the 3-4-5. (12 + 16)^2 = 784, which is much bigger than 400.