A bus is traveling from city A 10,
negative 10 to city B negative eight, eight. Its first stop is at C, which is
halfway between the cities. Its second stop is at D, which is
two-thirds of the way from A to B. What are the coordinates of C and D?
Well, the first thing we’ll have a
look at is stop C cause we can see that this is halfway between the cities. Well, if we’re looking to find the
midpoint between any two points, we have a formula to help us. And that formula is that the
coordinates of a midpoint is equal to, then we have 𝑥 sub one plus 𝑥 sub two over
two. So that is the 𝑥-coordinates of
both of our points added together then divided by two. And then for the 𝑦-coordinate, we
have 𝑦 sub one plus 𝑦 sub two over two, which is the 𝑦-coordinates added together
and then divided by two.
Well, now that we have the formula,
what we can do is use it to help us find the midpoint of our bus travel from city A
to city B, which is stop C. And so that we can achieve that, what we need to do is
label our coordinates, which we’ve done here. So we’ve got 𝑥 sub one, 𝑦 sub
one; 𝑥 sub two, 𝑦 sub two. So now, let’s substitute these into
the formula. And when we do that, we’re gonna
have that C, our midpoint, is going to be, then we’ve got for the 𝑥-coordinate 10
add negative eight because that’s 𝑥 sub one add 𝑥 sub two. And then this is divided by
two. And then for the 𝑦-coordinate, we
have negative 10 plus eight divided by two.
So when we calculate this, what
we’re going to get is two over two or two divided by two for our 𝑥-coordinate and
negative two over two for our 𝑦-coordinate. And it’s worth noting that we got
two over two or two divided by two for our 𝑥-coordinate because we had 10 and
negative eight. And if you add a negative, it’s the
same as subtracting a positive. So it’s the same as 10 minus eight,
which will give us our two. So when we do the calculation,
we’re gonna get one, negative one. So this is going to be the
coordinates of C, which is the halfway point between city A and city B.
So now, what we’re gonna have a
look at is point D. And we know that stop D or point D is two-thirds of the way from
A to B. Now, one way that we could work this out to find out the position of D would
be to find out what two-thirds of the distance of the 𝑥-coordinate from A to B is
and then two-thirds of the distance of the 𝑦-coordinate from A to B. And then add
this distance onto the original 𝑥- and 𝑦-coordinate of A. However, there is a
formal way that we could do that using a formula.
And the formula tells us that if we
want to find the point 𝑥, 𝑦, which is between two points, then this is equal to 𝑥
sub one plus 𝑘 multiplied by 𝑥 sub two minus 𝑥 sub one. Then for the 𝑦-coordinate, 𝑦 sub
one plus 𝑘 multiplied by 𝑦 sub two minus 𝑦 sub one, where 𝑘 is the fraction of
the total length or distance between the two points. Okay, great. So we have this formula. Let’s use it to help us find out
the position of point D.
So then we can say that point D is
at the point where we’ve got the 𝑥-coordinate with 10 plus two-thirds multiplied by
negative eight minus 10, where two-thirds is our 𝑘. Then for the 𝑦-coordinate, we’ve
got negative 10 plus two-thirds multiplied by eight minus negative 10. And two-thirds is our 𝑘 cause
we’re told that D is two-thirds of the way between A and B. So what we get for this,
for the 𝑥-coordinate, we’ve got 10 plus two-thirds of negative 18. Then for the 𝑦-coordinate, we’ve
got negative 10 plus two-thirds of 18. So this is gonna give us 10 minus
12 for the 𝑥-coordinate and negative 10 plus 12 for the 𝑦-coordinate.
So therefore, the final coordinates
for our point D are going to be negative two, two. So therefore, we can say the
coordinates of points C and D are one, negative one and negative two, two,