Video Transcript
In the opposite figure, the line
passing through the points 𝐴 and 𝐵 is parallel to the line passing through the
points 𝐶 and 𝐷. The line segment 𝐴𝐶 is parallel
to the line segment 𝐵𝐷. And the line segment 𝐹𝐺 is
perpendicular to the line segment 𝐴𝐵, where the length 𝐹𝐺 is equal to four
centimeters. If the area of the triangle 𝐴𝐵𝐸
is equal to 12 centimeters squared, find the length of the line segment 𝐶𝐷.
Let’s start with labeling the
diagram with the information given. The length of 𝐹𝐺 is four
centimeters, and the area of the triangle 𝐴𝐵𝐸 is 12 centimeters squared. Recall that the area of a triangle
is given by one-half times its base, 𝑏, times its perpendicular height, ℎ.
Looking at the triangle 𝐴𝐵𝐸, we
can take its base length to be the length of the line segment 𝐴𝐵. The height of the triangle is the
distance between the top line and the point 𝐸, which is equivalent to the distance
between the top and bottom lines. Since these lines are parallel and
the line segment 𝐹𝐺 is perpendicular to them, the distance between both lines is
just the length 𝐹𝐺. Therefore, the area of the triangle
is one-half times 𝐴𝐵 times 𝐹𝐺, which we are given as equal to 12 centimeters
squared.
We are also given that the length
𝐹𝐺 is equal to four centimeters. So on the left-hand side of this
equation, we have one-half times 𝐴𝐵 times four, which is equal to two 𝐴𝐵. So we have two 𝐴𝐵 equals 12. Dividing both sides by two gives us
the length of the line segment 𝐴𝐵, six centimeters.
Now consider the quadrilateral
𝐴𝐵𝐶𝐷. Since, as given in the question,
the line segments 𝐴𝐵 and 𝐶𝐷 are parallel, the line segments 𝐴𝐶 and 𝐵𝐷 are
also parallel. So this is a parallelogram, and the
parallel sides are of equal length. Therefore, the length of the line
segment 𝐶𝐷 is equal to the length of the line segment 𝐴𝐵, which is six
centimeters.
