Video Transcript
The given figure shows a
triangle 𝐴𝐵𝐶. (1) work out the value of
𝑥. And (2) work out the value of
𝑦.
We’re given a triangle with a
line parallel to one side inscribed within it and various lengths of segments of
the sides of the triangle. And we’re asked to find the
value of 𝑥 and the value of 𝑦.
Let’s begin with part (1),
which is finding 𝑥, where we see the two of our side segments involve 𝑥. We note first that a line of
length two units inside the triangle is parallel to the side 𝐵𝐶. Now, the side splitter theorem
tells us that this line divides the two sides 𝐴𝐶 and 𝐴𝐵 proportionally.
Remember, according to the side
splitter theorem, if a line parallel to one side of a triangle intersects the
other two sides of the triangle, then the line divides those sides
proportionally. If we label this line segment
𝐷𝐸 in our diagram, we could say that 𝐴𝐷 over 𝐷𝐵 is equal to 𝐴𝐸 over
𝐸𝐶. Substituting in the given
lengths, we can form an equation which we can solve for 𝑥. That’s three over two 𝑥 plus
three is equal to two over 𝑥 plus five. Now, multiplying both sides by
𝑥 plus five and two 𝑥 plus three and distributing the parentheses, we have
three 𝑥 plus 15 equals four 𝑥 plus six. Subtracting three 𝑥 and six
from both sides and swapping sides, we then have 𝑥 is equal to nine.
So now, making a note of this
and making some space, we can use this value of 𝑥 in part (2) of the question
to find the value of 𝑦. Now, since the two triangles
created by the intersection of side 𝐷𝐸 share a common angle 𝐴 and the pairs
of corresponding angles also created by this line are equal, we can say the
triangles 𝐴𝐷𝐸 and 𝐴𝐵𝐶 are similar triangles. In particular, this means that
the proportions 𝐴𝐷 over 𝐴𝐵 and 𝐷𝐸 over 𝐵𝐶 are equal. Now, we know that 𝐴𝐵 is equal
to the sum 𝐴𝐷 plus 𝐷𝐵. And that’s three plus two 𝑥
plus three. And we have 𝑥 equal to nine
from part (1). And so this evaluates to
24.
So now, substituting our values
𝐴𝐷 equals three, 𝐴𝐵 equals 24, 𝐷𝐸 equals two, and 𝐵𝐶 equals 𝑦 into our
equation, we have three over 24 is equal to two over 𝑦. Now, multiplying both sides by
𝑦 and by 24 over three, we have 𝑦 equal to 16. Hence, from the given figure,
we find that 𝑥 is equal to nine and 𝑦 is equal to 16.
