Question: Value of x

Comment on Value of x

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!
gmat-admin's picture

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 (, 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 using 2 equations and solving them for x
gmat-admin's picture

What 2 equations do you have in mind?

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

Why did you use the ratio above, instead of that?
gmat-admin's picture

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 Pythagorean theorem work here to find "x"? For example: x+1 squared plus 4x-3 squared = √3(x+1) squared?
gmat-admin's picture

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


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

Why did we use the ratio and not the enlargement factor? Is it because we have an equation on both sides?
gmat-admin's picture

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

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