In a treasure hunt, a sealed envelope was given to each team. The envelope contained a map drawn on a coordinate grid and the following information. The campsite is at the origin point; the campfire at five, four; the river at five, negative five; the tent at negative two, negative five; and a big tree at negative two, four. What is the distance between the campfire and the big tree?
There are also two other parts that we’ll come on to in a bit. So the first thing we’re gonna do is we’re gonna mark our points on a coordinate grid. So the first thing we’ve done to help us visualize is mark on our scale. Now we’re gonna mark on our points. So we have our campsite at the origin, so at the point zero, zero. We have our campfire at the point five, four; the river at five and negative five; the tent at negative two, negative five; then finally, the big tree at negative two, four. So great, we have all of our points marked off.
So now let’s answer the first question: what is the distance between the campfire and the big tree? We could see here we’ve drawn a line to represent the distance between the campfire and the big tree. Well, the first thing to notice about the campfire and the big tree is that they both have a 𝑦-coordinate at four. So therefore, what this means is, in fact, it’s a horizontal line between them. So we can just count how long this line is to work out the distance, which just counts as seven.
We could’ve also worked this out by working out the difference between the two 𝑥-coordinates. And to do that, what we would’ve had is the 𝑥-coordinate of the campfire, which is five, minus the 𝑥-coordinate of the big tree, which is negative two. And five minus negative two and that’s gonna give us seven. And that’s because if you have five minus negative two, well, if you just subtract a negative, it’s the same as adding a positive. So, that gives us the answer we’re looking for. Okay, great! So now let’s move on to the next part of the question.
Well, for the second part of the question, what we’re trying to do is find out what the distance is between the tent and the big tree. Well, what I’ve done is drawn the line again between the tent and the big tree. What we could see this time is that the 𝑥-coordinates are, in fact, the same. So therefore, it’s just the difference between our 𝑦-coordinates that’s gonna tell us the distance of our line, so the distance between the tent and the big tree.
So once again, we could’ve just done this by counting and worked out that it was nine. However, again, we could’ve worked out by working out the difference between, this time, the 𝑦-coordinates, so we’d have four minus negative five. So once again, if you subtract a negative, it’s the same as adding a positive. So four add five gives us nine, which is the value we got from counting. So now we can move on to the final part of the question.
So for the final part of the question we have, what is the distance between the campfire and the river?
Well, again, what we’ve done is drawn the line between the campfire and the river. Now, a good count, we could use methods we’ve used before because what we can see is that with campfire and the river, they’re both have the same 𝑥-coordinates. However, what we are gonna have a look at now is something called the distance between two points formula. And what the distance between two points formula tells us is the distance is equal to the square root of 𝑥 sub two minus 𝑥 sub one all squared plus 𝑦 sub two minus 𝑦 sub one all squared.
And what this is is the derivative of the Pythagorean theorem because what we’re saying is that we have our 𝑥-coordinates and the difference between them, which is gonna give us one of the sides of a right triangle. And then we’ve got our 𝑦-coordinates, and the difference between them would give us our other sides, so our vertical side of a right triangle. And then we square them and add them together. And that would give us the square of the longest side which’s our distance. So then what we would do is square root this to get what the distance itself is.
Okay, great! So we’re gonna apply this and see if this will work to find the distance between the campfire and the river. So what we’ve done first of all is labeled our coordinates, so we’ve got our 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. So our distance is gonna be equal to the square root of five minus five all squared plus negative five minus four all squared. So the distance is gonna be equal to the square root of zero squared plus negative nine squared, which is gonna be equal to the square root of 81. So this is gonna give us an answer of nine.
However, we could also get negative nine if we found the square root of 81. But we don’t use it in this context because we’re only interested in the magnitude, which is nine. Well, if we check on the drawing, we can count nine squares. We can see that the distance is gonna be nine between the campfire and the river. Also if we look across, it’s gonna be the same as the distance between the tent and the big tree because the two lines are the same length and parallel. So therefore, we’ve worked out each part of the question as seven, nine, and nine, respectively.