### Video Transcript

In the opposite figure, the line
passing through the points π΄ and π΅ is parallel to the line passing through the
points πΆ and π·. The line segment π΄πΆ is parallel
to the line segment π΅π·. And the line segment πΉπΊ is
perpendicular to the line segment π΄π΅, where the length πΉπΊ is equal to four
centimeters. If the area of the triangle π΄π΅πΈ
is equal to 12 centimeters squared, find the length of the line segment πΆπ·.

Letβs start with labeling the
diagram with the information given. The length of πΉπΊ is four
centimeters, and the area of the triangle π΄π΅πΈ is 12 centimeters squared. Recall that the area of a triangle
is given by one-half times its base, π, times its perpendicular height, β.

Looking at the triangle π΄π΅πΈ, we
can take its base length to be the length of the line segment π΄π΅. The height of the triangle is the
distance between the top line and the point πΈ, which is equivalent to the distance
between the top and bottom lines. Since these lines are parallel and
the line segment πΉπΊ is perpendicular to them, the distance between both lines is
just the length πΉπΊ. Therefore, the area of the triangle
is one-half times π΄π΅ times πΉπΊ, which we are given as equal to 12 centimeters
squared.

We are also given that the length
πΉπΊ is equal to four centimeters. So on the left-hand side of this
equation, we have one-half times π΄π΅ times four, which is equal to two π΄π΅. So we have two π΄π΅ equals 12. Dividing both sides by two gives us
the length of the line segment π΄π΅, six centimeters.

Now consider the quadrilateral
π΄π΅πΆπ·. Since, as given in the question,
the line segments π΄π΅ and πΆπ· are parallel, the line segments π΄πΆ and π΅π· are
also parallel. So this is a parallelogram, and the
parallel sides are of equal length. Therefore, the length of the line
segment πΆπ· is equal to the length of the line segment π΄π΅, which is six
centimeters.