Christmas conundrum answer/explanation

[NellyLunatic1980]

The conundrum, edited for content:

In an open field there are 5 Christmas trees decorated with colors. The red Christmas tree and the orange Christmas tree are 10 yards apart. The yellow Christmas tree is 12 yards from the red Christmas tree. The blue Christmas tree is 14 yards from the orange tree. The pink Christmas tree is 17 yards from the blue Christmas tree. The yellow Christmas tree and the Orange Christmas tree are 9 yards apart. The yellow Christmas tree and the pink Christmas tree are 13 yards apart. The pink Christmas tree and the orange Christmas tree are 11 yards apart. How far is the blue Christmas tree from the red Christmas tree (rounded to the nearest yard)?

Here is a diagram of all five trees, with each letter representing the tree whose color starts with that letter:



In order to find the distance between the red tree (R) and the blue tree (B), we need to find the measure of the angle ROB. We find that angle by finding the measures of angles ROY, YOP, and POB, then subtracting that total from 360°. We find these angles through the "cosine rule", which states that for any triangle whose sides measure a, b, and c:

a² = b² + c² - 2bc cos θ, where θ is the angle that is opposite of side "a"

Since we are looking for the angle, we shuffle this equation so that we can solve for θ:

θ = cosˉ¹ [(b² + c² - a²) / 2bc]

For triangle ROY, if a=12, b=10, and c=9:

θ(ROY) = cosˉ¹ [(10² + 9² - 12²) / 2*10*9]
θ(ROY) = cosˉ¹ [(100 + 81 - 144) / 180]
θ(ROY) = cosˉ¹ (37/180)
θ(ROY) = cosˉ¹ .205555555...
θ(ROY) = 78.14°

For triangle YOP, if a=13, b=11, and c=9:

θ(YOP) = cosˉ¹ [(11² + 9² - 13²) / 2*11*9]
θ(YOP) = cosˉ¹ [(121 + 81 - 169) / 198]
θ(YOP) = cosˉ¹ (33/198)
θ(YOP) = cosˉ¹ .166666666...
θ(YOP) = 80.41°

For triangle POB, if a=17, b=14, and c=11:

θ(POB) = cosˉ¹ [(14² + 11² - 17²) / 2*14*11]
θ(POB) = cosˉ¹ [(196 + 121 - 289) / 308]
θ(POB) = cosˉ¹ (28/308)
θ(POB) = cosˉ¹ .0909090909...
θ(POB) = 84.78°

Now we can solve for angle ROB:

θ(ROB) = 360° - 78.14° - 80.41° - 84.78° = 116.67°

Finally, for triangle ROB, if b=14, c=10, and θ(ROB)=116.67°, we solve for "a", which is the answer to the conundrum:

a² = b² + c² - 2bc cos θ
a² = 14² + 10² - (2*14*10)(cos 116.67°)
a² = 196 + 100 - (280)(cos 116.67°)
a² = 296 - (280)(-.44885)
a² = 296 - (-125.678)
a² = 421.678
a = 20.534799 = 21 yards

[andrewjackson]

Thanks but doesn't it change RB if OP is running the other way.

Or if OB goes up toward Y.

There should be another one but I can't remember what has to change.

I'll see if I can post the alternate diagrams tonight.

[Greyhound Dude]

Or you could have drawn it in CAD software like I did in less than 30 seconds and come up with the answer. Who needs math anyways.

[Bob78164]

Here is a diagram of all five trees, with each letter representing the tree whose color starts with that letter:



In order to find the distance between the red tree (R) and the blue tree (B), we need to find the measure of the angle ROB. We find that angle by finding the measures of angles ROY, YOP, and POB, then subtracting that total from 360°. We find these angles through the "cosine rule", which states that for any triangle whose sides measure a, b, and c:

a² = b² + c² - 2bc cos θ, where θ is the angle that is opposite of side "a"

Since we are looking for the angle, we shuffle this equation so that we can solve for θ:

θ = cosˉ¹ [(b² + c² - a²) / 2bc]

For triangle ROY, if a=12, b=10, and c=9:

θ(ROY) = cosˉ¹ [(10² + 9² - 12²) / 2*10*9]
θ(ROY) = cosˉ¹ [(100 + 81 - 144) / 180]
θ(ROY) = cosˉ¹ (37/180)
θ(ROY) = cosˉ¹ .205555555...
θ(ROY) = 78.14°

For triangle YOP, if a=13, b=11, and c=9:

θ(YOP) = cosˉ¹ [(11² + 9² - 13²) / 2*11*9]
θ(YOP) = cosˉ¹ [(121 + 81 - 169) / 198]
θ(YOP) = cosˉ¹ (33/198)
θ(YOP) = cosˉ¹ .166666666...
θ(YOP) = 80.41°

For triangle POB, if a=17, b=14, and c=11:

θ(POB) = cosˉ¹ [(14² + 11² - 17²) / 2*14*11]
θ(POB) = cosˉ¹ [(196 + 121 - 289) / 308]
θ(POB) = cosˉ¹ (28/308)
θ(POB) = cosˉ¹ .0909090909...
θ(POB) = 84.78°

Now we can solve for angle ROB:

θ(ROB) = 360° - 78.14° - 80.41° - 84.78° = 116.67°

Finally, for triangle ROB, if b=14, c=10, and θ(ROB)=116.67°, we solve for "a", which is the answer to the conundrum:

a² = b² + c² - 2bc cos θ
a² = 14² + 10² - (2*14*10)(cos 116.67°)
a² = 196 + 100 - (280)(cos 116.67°)
a² = 296 - (280)(-.44885)
a² = 296 - (-125.678)
a² = 421.678
a = 20.534799 = 21 yards

What happens if you fold your diagram at the line OP, moving point B to the other side of that line? What happens if you fold your diagram at line OY, moving point R to the other side of that line? What happens if you do both? Don't you get three different answers for the distance from R to B?

In other words, you can't prove (because it's not necessarily true) that the four angles you're measuring necessarily add up to 360°. --Bob

[andrewjackson]

OK, I sat down and did this all over again. I still get four different distances. But only three different answers once I round to the nearest yard because I made a rounding error before.

If the Red tree is near the Pink tree, then it is 17 yards away from the Blue tree no matter if the Blue tree is near the Yellow tree or not.

If the Red tree is on the other side of the Orange tree from the Pink tree then you get two different answers depending on the location of the Blue tree. If the Blue tree is near the Yellow tree then it is 16 yards away from the Red tree. If the Blue tree is on the other side of the Orange tree and far away from the Yellow tree then it is 21 yards from the Red tree.

So the answers are 17, 17, 16, and 21 depending on the particular location of the Red and Blue trees to the other three.

I just don't know how to prove it.

OK, one way might work. Look at Nelly's diagram.



Imagine that he had drawn the Red tree near the Pink one. The distances from the Orange and Yellow to the Red are just one yard different than the distances from the Orange and Yellow to the Pink, right?

So the Red tree could be just about 1 yard away from the Pink tree, right? How in the world could it be 21 yards away from the Blue one if the Pink tree is only 17 yards away from the Blue one. That's because it isn't. It would actually be a little more than 16.5 yards away.

I'm now satisfied even if everyone else isn't. There are multiple answers to this question even if the oak tree is actually the orange one.