Video Transcript
Find the length of line segment
𝐴𝐷 and line segment 𝐷𝐶.
We can observe in the figure that
we have two triangles, 𝐴𝐵𝐶 colored here in orange and 𝐷𝐶𝐸 colored in pink. We need to find the lengths of line
segments 𝐴𝐷 and 𝐷𝐶.
Now, we might observe that we have
some congruent side and angle measures in these two triangles. So it might be sensible to begin by
checking if the two triangles are congruent. Firstly, we have two angles marked
as having a measure of 68 degrees. These are the measures of angles
𝐴𝐵𝐶 and 𝐷𝐶𝐸.
Then, we have another pair of
congruent angles, since angles 𝐴𝐶𝐵 and 𝐷𝐸𝐶 both have measures of 31
degrees. And we also have two sides that are
congruent. Line segments 𝐵𝐶 and 𝐶𝐸 are
both given as 5.7 centimeters. So, we have two pairs of congruent
angles. And importantly, the congruent pair
of sides are the included sides between the two pairs of angles. This allows us to apply the ASA, or
angle-side-angle congruence criterion, to prove that triangles 𝐴𝐵𝐶 and 𝐷𝐶𝐸 are
congruent. Knowing that the triangles are
congruent will then allow us to find out more length information.
Let’s see if we can find the length
of the line segment 𝐴𝐷. Line segment 𝐴𝐷 forms part of the
line segment 𝐴𝐶 in triangle 𝐴𝐵𝐶. And line segment 𝐴𝐶 will have a
corresponding side length in triangle 𝐷𝐶𝐸. It’s the line segment 𝐷𝐸. In congruent triangles,
corresponding side lengths are equal. So we know that the length of 𝐴𝐶
is the same as 𝐷𝐸, which is 5.4 centimeters.
But we aren’t asked for the length
of 𝐴𝐶, we are asked for the lengths of 𝐴𝐷 and 𝐷𝐶. So let’s consider line segment
𝐷𝐶. It corresponds to line segment 𝐴𝐵
in triangle 𝐴𝐵𝐶. And so, both of these lengths are
three centimeters. We can work out the remaining
length, which is of line segment 𝐴𝐷, by subtracting three centimeters from 5.4
centimeters, which gives us 2.4 centimeters.
We can therefore give the answer
that the length of line segment 𝐴𝐷 is 2.4 centimeters and the length of line
segment 𝐷𝐶 is three centimeters.
