Question: Value of x

Comment on Value of x

I solved as : since triangle is 30-60-90 triangle, so the lengths are in the ratio - x : sq root 3 *x : 2*x

Now 2x corresponds to side of length 4x -3, so I took
2x= 4x -3 . when solved for x, I get x = 3/2. Please correct me as to where I'm wrong?
greenlight-admin's picture

In your equation, 2x = 4x -3, the x on the LEFT side does not represent the same value that x represents on the RIGHT side. So, your equation isn't valid.

Notice that, the two lengths (x+1) and (4x-3) could also be (x+3) and (4x). If we apply your strategy here, we conclude that 2x = 4x, which makes no sense.

Instead, we must us the RELATIONSHIPS between corresponding sides to solve this question.

Thank you Brent. I have one more question. Since we have variables on both sides of the equation, so they must be equal to produce appropriate result. But incase we have integers instead of variables for lengths, in that case can I use my approach? say for example I have hypotenuse of length 6 in 30-60-90 triangle, then can I equate it as 2x =6 so x=3. Will that help in solving faster and will it still be correct? please suggest.
greenlight-admin's picture

Sorry, but I'm not sure what you're asking.
If you have a 30-60-90 triangle, and the hypotenuse has length 6, then you can conclude that the shorter leg has length 3 and the other leg has length 3√3

Does that help?

Do you get the comments if typed here?
greenlight-admin's picture

Yes, I see the comments here.

Hello Brent, I clicked on the reply button and asked my doubt again. Seems like it has not been delivered to you, has it?
greenlight-admin's picture

Yes, I received the reply.
I didn't get back to you immediately, but I always answer questions within 24 hours.


Can we also use the enlargement factor here or is it solely used when given one side of the length
greenlight-admin's picture

We can certainly apply the enlargement factor here.

On the shown triangle, the side with length x+1 corresponds to the side with length 1 in the BASE triangle.

So, the enlargement factor = (x+1)/1 = x+1

On the shown triangle, the side with length 4x-3 corresponds to the side with length 2 in the BASE triangle.

So, we can write: (2)(x+1) = 4x-3
Expand: 2x + 2 = 4x - 3
Solve, to get: x = 5/2


Trigonometry clicked me here , sin 60 = opposite/hypotenuse
greenlight-admin's picture

If you remember that sin 60 = √3/2, then that will help you.
However, trigonometric ratios are beyond the scope of the GRE, so you can also just apply what you know about 30-60-90 right triangles.


Since, as you've pointed out in previous lessons, the 90˚ side is always twice the side opposite 30˚, for a 30 60 90 triangle, we could say:

2(side opposite 30˚) = side opposite 90˚
2(x+1) = 4x-3

and solve for x.

Not saying it's better, just another approach.

greenlight-admin's picture

That's a great approach! In fact, it shaves a step off my solution!

Using similar triangles, I first wrote: (x+1)/1 = (4x-3)/2
After cross multiplying, I got: 2(x+1) = 4x-3

In your (faster) solution, you started with 2(x+1) = 4x-3

Very nice!


Hi Brent, for this problem could we find the third liters in terms of x as (x+1)rt3 and then use py theorem to solve for x. It’s a bit of a longer approach but wanted to know if it works?
greenlight-admin's picture

That approach will also work (although you end up with a pretty tricky quadratic equation to solve)

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