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Comment on Value of x
For the 30:60:90 Triangle,
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
What 2 equations do you have
What 2 equations do you have in mind?
I got the correct answer
Why did you use the ratio above, instead of that?
Your ratios work as well.
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
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
Hi Brent, in 30-60-90 right
Sorry, I don't follow your
Sorry, I don't follow your question. Can you rephrase it?
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
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
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
Why did we use the ratio and
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.
Could you please solve this
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!
Hi Brent, could the
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
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,
For this question my brain is just dumb until I realized two sides are equal doesn't mean the other two sides are equal .
Question link: https:/
Question link: https://gmatclub.com/forum/is-quadrilateral-pqrs-a-parallelogram-1-adjac...
It's better to learn that property while practicing than during the test :-)
actually there is a theory
That's correct. The side
That's correct. The side opposite the 30 degree angle is half as long as the hypotenuse.