### Video Transcript

In the figure, what is the length
of line segment πΈπΊ?

In the diagram, we can observe that
we have the triangle π·πΈπΊ, which has the line π·πΉ passing through it. Given that we have two line
segments that are both given as 10.5 length units, then we know that these sides are
congruent. And by definition, that means that
triangle π·πΈπΊ is an isosceles triangle.

Now, letβs consider the line
π·πΉ. From the markings on the diagram,
we notice that line π·πΉ, which passes through the vertex π·, meets the base of line
segment πΈπΊ at 90 degrees. And we can recall an important
theorem regarding these properties.

It is that the straight line that
passes through the vertex angle of an isosceles triangle and is perpendicular to the
base bisects the base and the vertex angle. So, because the angle at the vertex
is bisected, then the measure of angle πΊπ·πΉ equals the measure of angle
πΈπ·πΉ.

And importantly for this question,
we know that the base is bisected. So the line segment πΈπΉ has the
same length as line segment πΊπΉ, which is 6.5 length units.

We are asked to find the length of
the line segment πΈπΊ, which we can find by adding the two lengths on the base. And 6.5 plus 6.5 gives us the
answer that the length of line segment πΈπΊ is 13, or 13 length units.