Given the information in the
diagram below, if the vector 𝐁𝐂 is equal to 𝑘 times the vector 𝐄𝐃, find 𝑘.
In our diagram, we’ve actually been
given two similar triangles. That is, one is an enlargement of
the other. Triangle 𝐀𝐃𝐄 has been enlarged
onto 𝐀𝐁𝐂. Now, we know this since angle 𝐴
here is a shared angle. We see that angle 𝐴𝐸𝐷 and 𝐴𝐶𝐵
are equal as are angles 𝐴𝐷𝐸 and 𝐴𝐵𝐶. And this is because corresponding
angles are equal and we know sides 𝐸𝐷 and 𝐶𝐵 are parallel.
Since the triangles have equal
angles, they must be similar. And this means we’re able to
calculate a scale factor of enlargement. This is found by dividing the
length of the enlarged triangle by the corresponding length of the original. We sometimes write this as new
length divided by old length.
If we take the enlarged triangle to
be 𝐴𝐵𝐶 and the original triangle to be 𝐴𝐷𝐸, we see that we can find the scale
factor by dividing the length 𝐴𝐵 by the length 𝐴𝐷. The length of 𝐴𝐵 is actually the
sum of the two dimensions given. It’s 7.8 plus 5.2, which is 13
centimeters. And so the scale factor for
enlargement here is 13 divided by 5.2, which is five over two.
Now, the scale factor is simply a
multiplier. We know that to enlarge triangle
𝐴𝐷𝐸 onto 𝐴𝐵𝐶, we’d multiply any of its lengths by the scale factor of five
over two. Now, we’re trying to find a
relationship between the vectors 𝐁𝐂 and 𝐄𝐃. Well, we know that the dimension
𝐶𝐵 must be five over two times the dimension 𝐸𝐷. But since these lines are also
parallel, we know that they have the same direction. And this means we can say that the
vector 𝐂𝐁 must be five over two times the vector 𝐄𝐃.
The problem is, we want to work out
the vector 𝐁𝐂 in terms of the vector 𝐄𝐃. Currently, we have the vector 𝐂𝐁
in terms of 𝐄𝐃. And so we’re traveling in the
opposite direction. And we remember that to do this
with vectors, we change the sign so that the vector 𝐁𝐂 is equal to the negative
vector 𝐂𝐁. We just showed that the vector 𝐂𝐁
is five over two times the vector 𝐄𝐃. So this must mean that the vector
𝐁𝐂 is negative five over two times the vector 𝐄𝐃. Comparing this to the original form
in the question, and we see then that 𝑘 is equal to negative five over two.