Geometry (4th Ed) Guide Ch 6 Question 8

[howardsun]

For this question, I'm trying to understand the calculation.

The side of an equilateral triangle has the same length as the diagonal of a square. What is the area of the square?

1. The height of the equilateral triangle is equal to 6*sqrt(3)
2. The area of the equilateral triangle is equal to 36*sqrt(3)

For statement one, I know from the height 6*sqrt(3), and the ratio being x:x sqrt(3):2x, that the x is 6 and length is 12 for all sides, that plugging the 12=s*sqrt(2) I can solve for base and height of the square. -- Sufficient

For statement two, I'm stuck. Since I have 36*sqrt(3), I can double that to get B*H of 72*sqrt(3), from there I'm a bit lost on how the answer in the guide solved from 72*sqrt(3) into 12 and 6*sqrt(3) to get sufficiency.

Could someone give me a step by step in going from 72*sqrt(3) to the answer?

[RonPurewal]

weeeeeelllll, the most important thing is that you shouldn't be doing ANY calculations in this problem.

remember what the goal is (and isn't)!
the goal in this problem is just to determine whether there's (a) just one solution, or (b) more than one solution.
that's it -- you don't care a whit about the numerical value(s) of the solution(s).

so, here's all the reasoning you need here:

STATEMENT 1
* You have the height of the equilateral triangle.
* This means you know everything about the triangle (since all equilateral triangles have the same shape).
* So you've got the diagonal of the square.
* That means you know everything about the square (since all squares have the same shape).
Done. Sufficient.

STATEMENT 2

* You have the area of the equilateral triangle.
* This means you know the size of the triangle ... and so you know everything about the triangle (since all equilateral triangles have the same shape).
* So you've got the diagonal of the square.
* That means you know everything about the square (since all squares have the same shape).
Done. Sufficient.

this is the most important lesson to learn in this specific problem -- the fact that you shouldn't even be doing the algebra in the first place.

[RonPurewal]

still, in terms of reviewing this problem (for things you can use in the future), there's value in knowing how to do the calculations.
(again, this is not the same thing as thinking that the calculations are valuable in this problem; in this problem, the calculations are a waste of time.)

so, if you know the area of the triangle...
... let the base of the triangle be "B"
... then half the base is B/2
... using the proportion that you already understand (you used it above), the height is (B/2) x √3 = B√3/2
... so, the area is
(1/2)(base)(height)
= (1/2)(B)(B√3/2)
= (B^2)√3/4.

then you can set (B^2)√3/4 equal to the given area and find the value of B.