Question: Value of x

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.

What 2 equations do you have in mind?

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.

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