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Wizard
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Wizard
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January 26th, 2020 at 7:18:43 AM permalink
Given the larger figure is a square, solve for x.



Usual rules apply:

1. No searching
2. No posting links to solutions elsewhere.
3. Please put answers and solutions in spoiler tags until I announce a satisfactory solution.
4. Beer to the first satisfactory answer and solution.
5. Those who have won a beer in the last year may not play for the 24 hours following this posting.
Last edited by: Wizard on Jan 26, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
rsactuary
rsactuary
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January 26th, 2020 at 7:33:37 AM permalink
Is the yellow triangle an equilateral triangle? or does that have to be determined?
OnceDear
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OnceDear
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January 26th, 2020 at 7:56:17 AM permalink
Quote: rsactuary

Is the yellow triangle an equilateral triangle? or does that have to be determined?

No. It can't be.
If you are enjoying the game, you're already winning.
OnceDear
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OnceDear
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January 26th, 2020 at 8:44:30 AM permalink
GGgggaaahhhhh.
I'm doing it, but somebody is sure to beat me to it.
If you are enjoying the game, you're already winning.
ThatDonGuy
ThatDonGuy
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January 26th, 2020 at 9:21:48 AM permalink
Quote: Wizard

This post was removed in the interests of giving others who haven't won a beer a chance at it. I copied and pasted the text to Don. Those who haven't won a beer yet have until 24 hours since the OP to solve it.

Hopefully nobody will cheat who may remember Don's post.

Last edited by: unnamed administrator on Jan 26, 2020
Wizard
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Wizard
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January 26th, 2020 at 9:52:08 AM permalink
Quote: rsactuary

Is the yellow triangle an equilateral triangle? or does that have to be determined?



It has to be determined.
It's not whether you win or lose; it's whether or not you had a good bet.
BigJer
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Thanks for this post from:
odiousgambit
January 26th, 2020 at 1:23:52 PM permalink
Lol. Are the numbers the area or a side of the triangles?
The Terror of Casinos.
rsactuary
rsactuary
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Thanks for this post from:
BigJer
January 26th, 2020 at 1:43:10 PM permalink
Quote: BigJer

Lol. Are the numbers the area or a side of the triangles?



I think it has to be area.
Joeman
Joeman 
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January 26th, 2020 at 2:19:25 PM permalink
15.49


Let's say a side of the square has length L. The value of x is equal to the square area minus that of the triangles, or:

x = L^2 - 20

So, the key is solving for L^2, which we can get by solving a set of equations for the areas of the three triangles. The blue and green triangles both have a height of L. We can call their respective bases b1 and b2.

1/2 L x b1 = 8
1/2 L x b2 = 5

Solving each for the base in terms of L, we get:

b1 = 16/L
b2 = 10/L

The base and height of the red triangle would be L - b1 and L - b2. So, its area can be represented by:

1/2 (L - b1) x (L - b2) = 7 or
L^2 - Lb1 - Lb2 + 10/L x 16/L = 14

Substituting from above, we get:

L^2 - L(16/L) - L(10/L) + (16/L)(10/L) = 14

Which, after reducing, rearranging, and finally multiplying all terms by L^2, we get:

L^4 - 40L^2 + 160 = 0

Using the quadratic equation**, we can solve for L^2, which would be an area for the square of 35.49. Subtracting out the area of the known triangles, we get x = 15.49.

** FULL DISCLOSURE: I mis-remembered the formula for the quadratic equation (it's been quite a while...), and when my first answer didn't make sense, I had to look up the formula. Not sure if this would disqualify me for the beer.
"Dealer has 'rock'... Pay 'paper!'"
OnceDear
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OnceDear
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January 26th, 2020 at 2:30:28 PM permalink
I think I have it

edit, I got my labelling wrong when I scribbled it out. correcting...
Let base side of square be y
Let short redgreen side be g
Let short blue side be b
Let short green side be g1 red side be r1
Let long green side be g2 red side be r2 (Not hypoteneuse)


Using area of triangle = 1/2 base x height

5=g.y/2
y=10/g

8=b.y/2
y=16/b

Looking at red triangle of area =7=r1.r2/2
14=r1.r2
but r1=y-b and r2=y-g
14=(y-b)(y-g)
14=(y-16/y).(y-10/y)
14=y^2-16-10+160/y^2
40=y^2+160/y^2
Now y^2 is the area of the square. Call y^2 S
0=S+160/S-40
multiply both sides by S
0=S^2-40S+160

a simple quadratic with solutions
S=35.49193338483
or
S=4.5080666151703
Now we reject the second answer because the total area is at least the sum of 8,5,7
So total square area is 35.49
therefore area x=35.49-8-7-5=15.49


I'm off to check my workings now. Just wanted to get it posted
Last edited by: OnceDear on Jan 26, 2020
If you are enjoying the game, you're already winning.

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