In the figure below, triangle ADE is an equilateral triangle. If DC=AE+BE and AC=3 inches, what is the length of AB?a. 2.5b. 2c. 1.5d. 1e. None of these.

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vixen999 | High School Teacher | (Level 2) Adjunct Educator

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There might possibly be a better way to solve it, but here is one approach. 

Let's let AE = x, and BE = y.

AE + EB = AB (Segment Addition Postulate)


So, by substitution, we get

AE + EB = DC and x + y = AB (substitution)

So, AC = AD + DC (Segment Addition Postulate)

and then, AC = x + x + y, or 3 = 2x + y

Then, we can plug in numbers for x and see what works.

If x = 2.5, then we get 3 = 5 + y and y would have to be negative which is not possible with a shape.

If x = 2, then we get 3 = 4 +y and again we would have a negative number for y which is not possible.

If x = 1.5, then we get 3 = 3 + y and y would have to be 0 which is not possible in a triangle.

If x = 1, then we get 3 = 2 + y so y would be 1.  As long as it is OK to have x and y be equal to the same number, then the answer is 1.

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