# Question: Value of x

## Comment on Value of x

### For the 30:60:90 Triangle,

For the 30:60:90 Triangle, does it matter which leg we use for 1 and √3? In your base triangle,√3 is on the base. Could is also work if √3 is on the other leg? Thanks! ### The 3 sides in a 30:60:90

The 3 sides in a 30:60:90 triangle have lengths 1, √3, and 2, and there is a strict rule for which sides have which length.

In an earlier lesson (https://www.gmatprepnow.com/module/gmat-geometry/video/860), we learned that, in a triangle, the longest side is the side opposite the biggest angle, and the shortest side is the side opposite the smallest angle.

So, in a 30:60:90 triangle, the shortest side is opposite the smallest angle (30 degrees). So, the side opposite the 30-degree angle must have length 1.

Likewise, the side opposite the 60-degree angle must have length √3, and the side opposite the 90-degree angle must have length 2.

### is there a way to solve this

is there a way to solve this using 2 equations and solving them for x ### What 2 equations do you have

What 2 equations do you have in mind?

### I got the correct answer

I got the correct answer using ratio 1/2 = (x+1)/(4x-3)

Why did you use the ratio above, instead of that? ### Your ratios work as well.

You're comparing the shortest side (x+1) and the hypotenuse (4x-3) on the LARGER triangle with the shortest side (1) and the hypotenuse (2) on the SMALLER triangle.

I'm comparing corresponding sides.

As long as you keep the order straight, either solution is fine.

### Just an idea, but could the

Just an idea, but could the Pythagorean theorem work here to find "x"? For example: x+1 squared plus 4x-3 squared = √3(x+1) squared? ### Great idea. We know that, in

Great idea. We know that, in the 30-60-90 right triangle, the length of the missing side will be √3 times the length of the side opposite the 30-degree angle (x+1)

So, the length of the missing side is √3(x+1)

Let's keep going.

First, we'll apply the Pythagorean theorem...
We get: (x + 1)² + [√3(x+1)]² = (4x - 3)²
Expand: x² + 2x + 1 + 3x² + 6x + 3 = 16x² - 24x + 9
Simplify: 4x²+ 8x + 4 = 16x² - 24x + 9
Rearrange to get: 12x² - 32x + 5 = 0
Factor (ugh!) to get: (6x - 1)(2x - 5)
Solve: EITHER x = 1/6 OR x = 5/2

Notice that, if x = 1/6, then the length of the hypotenuse (4x - 3) is NEGATIVE (which makes no sense).
So, we can ELIMINATE the solution x = 1/6, which means the correct answer is x = 5/2

So the Pythagorean Theorem works here. The only problem is that we must content with the tricky quadratic equation, 12x² - 32x + 5 = 0

Cheers,
Brent

### Hi Brent, in 30-60-90 right

Hi Brent, in 30-60-90 right triangle, so we have x:√3x:2x. Here x = (x + 1). Therefore not quite sure why (4x - 3) not * (x + 1) here but only √3 ? Thanks Brent. ### Sure Brent.

Sure Brent.

In 30-60-90 right triangle, so we have x : √3x : 2x.
Here x = (x + 1). Therefore not quite sure why (4x - 3) not multiply with (x + 1) here but only √3 in below calculation of using Pythagorean theorem ? Thanks Brent.

(x + 1)² + [√3(x+1)]² = (4x - 3)² ### Sorry, I'm having a hard time

Sorry, I'm having a hard time understanding your original question, and your follow-up question.

If we use (x+1) has the enlargement factor (which is fine), then the hypotenuse will have length (2)(x+1)
Since 4x-3 is the given length of the hypotenuse we can say: (2)(x+1) = 4x-3
When we solve this equation we get x = 5/2

### Sorry Brent. Let me copy your

Sorry Brent. Let me copy your original post as below:

In the 30-60-90 right triangle, the length of the missing side will be √3 times the length of the side opposite the 30-degree angle (x+1)

So, the length of the missing side is √3(x+1).

My question is why is √3 times the length of the side opposite the 30-degree angle (x+1) only? What about the other two sides? Hope it's clear now? Thanks Brent ### Ahhh.

Ahhh.
Notice that the ratio of the sides in a 30-60-90 right triangle are x : (√3)x : 2x
Notice that the side opposite the 60 degree angle, (√3)x is √3 times the length of the side opposite the 30 degree angle.

So if the side opposite the 30 degree angle has length x+1, then the side opposite the 60 degree angle will have length (√3)(x+1)

### OUCh! I think I'll stick with

OUCh! I think I'll stick with your method.

### Why did we use the ratio and

Why did we use the ratio and not the enlargement factor? Is it because we have an equation on both sides? ### We can use ratios or the

We can use ratios or the enlargement factor to find other lengths.
In actuality, the two techniques are practically identical.
Here's what I mean:
Let's say, we have two similar triangles, and two corresponding sides have lengths 1 and 3 respectively.
Two other corresponding sides have lengths 2.5 and x respectively.

From the first piece of information, we can automatically see that the enlargement factor is 3, which means x = (3)(2.5) = 7.5
Alternatively, we can use ratios to write: 1/3 = 2.5/x
When we cross multiply and solve, we get x = 7.5

So, either approach will work.

Cheers, Brent

### Could you please solve this

Could you please solve this question using the enlargement factor? Thanks :) ### You bet.

You bet.

In the BASE 30-60-90 triangle, the side opposite the 30° has length 1.
In the GIVEN 30-60-90 triangle, the side opposite the 30° has length x+1.
(x+1)/1 = x+1
So, the enlargement factor = x+1
In other words, the GIVEN triangle is (x+1) times the size of the BASE triangle.

In the BASE 30-60-90 triangle, the hypotenenuse has length 2.
So, the hypotenuse length of the GIVEN triangle = (2)(x+1)

Since we're told the hypotenuse of the GIVEN triangle has length 4x-3,
we can write: (2)(x+1) = 4x-3
Expand: 2x + 2 = 4x - 3
Add 3 to both sides: 2x + 5 = 4x
Subtract 2x from both sides: 5 = 2x
Solve: x = 5/2

### You rock! ;-) Thanks!

You rock! ;-) Thanks!

### Hi Brent, could the

Hi Brent, could the enlargement factor = 4x-3 too? If yes, could you show the calculations as I got stuck? Thanks ### No, we can't say that the

No, we can't say that the enlargement factor is 4x-3. Here's why:
In our base 30-60-90 right triangle, the hypotenuse has length 2.
If the enlargement factor were, indeed, 4x-3, the length of the hypotenuse in the given 30-60-90 right triangle would equal (2)(4x-3), which simplifies to be 8x-6.
However, in the given 30-60-90 right triangle, the hypotenuse has length 4x-3 (not 8x-6).

### Thanks Brent. This is a very

Thanks Brent. This is a very tricky one. I thought if 4x-3 were used as enlargement factor, it will be (4x-3)/2? ### Be careful. I said that 4x-3

Be careful. I said that 4x-3 is not the enlargement factor.

However, if we want to compare the two hypotenuses, we COULD say that the enlargement factor is (4x-3)/2, since 4x-3 is the length of the hypotenuse of the given 30-60-90 right triangle, and 2 is the length of the hypotenuse of the base 30-60-90 right triangle, That's correct.

### https://gmatclub.com/forum/is

For this question my brain is just dumb until I realized two sides are equal doesn't mean the other two sides are equal .  