Geometry book, Hard Practice Set, Question 15

[DanC]

There are four 30-60-90 triangles in this problem, and following the 1:rt3:2 method I got RT=1/4 before solving for the other sides of the RST triangle, which is where we find RS, which is the quantity we must solve for in the question. With RT=1/4, it seems to me RS = 1/(4rt3). That's because RT = 1/4 divided by rt3, which becomes 1/4 times 1/rt3.

But the final line in the answer explanation does something different. Why multiply by rt3/rt3 instead of 1/rt3?

[DanC]

Another way to solve the problem is to subtract QR from QS to find RS.

QS=1/rt3
QR=rt3/4

(1/rt3)-(rt3/4) = (4/4rt3)-(3/4rt3) = 1/4rt3

Why is that not the right answer?

[tommywallach]

Dan,

As I said on the other question, you never leave radicals in the denominator. And you can't just multiply something by 1/rt. 3--that would change the number. We multiply by rt. 3/rt. 3 because that's the same as multiplying by 1, which doesn't change the actual number, just the way we're expressing it.

And yes, your other method works great too, you just then have to multiply by rt. 3/rt. 3.

-t

[DanC]

If I have 1/4 divided by rt3, isn't that the same as 1/4 times 1/rt3?

That's why I did that. I wasn't suggesting that you can "just multiply something by 1/rt3". In fact, that step is necessary to find the answer, and is not exclusive of the multiplication by rt3/rt3 step.

Does the book actually state the purpose behind that final multiplication? I don't recall seeing that explanation in the book, so thanks for clarifying.

[tommywallach]

Aha! I understand now. Sorry for the confusion.

As for stating the reason, I'm not sure if it's in that book. It's more likely to be stated in the algebra book, somewhere or other. : )

-t