The boys mixed 10 gallons of the 20% pure lemon juice and 10 gallons of the 70% pure lemon juice mix, because they wanted 20 gallons of lemonade at a 40% lemon juice mixture. how much should they have used to get a final mixture of 20 gallons at 40% lemon juice?

The boys should use 140% of pure lemon juice mi to get a final mixture of 20 gallons at 40% lemon juice:

if 20:70 the 40:70 - it is more like ration and proportion. 

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This is an isosceles triangle. The definition of an isosceles triangle is a triangle with at least two congruent sides and angles. If 2 angles on a triangle are congruent (in this case 45 and 45 are two congruent angles) then triangle is isosceles. Therefore the two sides of triangle will be congruent. We know that the triangle is a right triangle because it has a hypotenuse. If a triangle has a hypotenuse then it's a right triangle. We can apply the Pythagorean theorem: a^2 + b^2 = c^2
A and B are the legs and C is the hypotenuse.
We can plug C in the equation:
a^2 + b^2 = 128
What do we know about the legs of the isosceles triangle? They are congruent so a and b have to be equal. From here it's simply guess and check. Will 8 work?
8^2 + 8^2 = 128
64 + 64 = 128
128=128
Yes the value 8 works so the length of two legs of the triangle is 8.