### Video Transcript

Given that π΄π΅πΆπ· is a trapezoid, determine the length of line segment π·πΆ.

Weβre told that the quadrilateral π΄π΅πΆπ· is a trapezoid, which means it has one
pair of parallel sides. We can see these parallel sides marked on the figure. Theyβre the sides π΄π· and π΅πΆ. Weβre asked to determine the length of the line segment π·πΆ, which is one of the
nonparallel sides of the trapezoid, also known as its legs.

Now, it may appear at first that we donβt have enough information to be able to do
this. However, if we sketch in the perpendicular from point π· to the side π΅πΆ, meeting
this side at a point weβll call πΈ, then we create a right triangle in which π·πΆ is
the hypotenuse. The length of this new line segment is the same as the length of line segment π΄π΅,
as both are perpendicular to the two parallel sides of the trapezoid. Hence, π·πΈ equals 12 centimeters.

We can also determine the length of line segment πΈπΆ. As π΅ is vertically below π΄ and πΈ is vertically below π·, line segments π΄π· and
π΅πΈ are of equal length. Hence, π΅πΈ is also 12.1 centimeters. And the length of πΈπΆ is the difference between the lengths of π΅πΆ and π΅πΈ. Thatβs 21.1 minus 12.1, which is nine centimeters.

We now have a right triangle in which we know the lengths of two of the sides and
wish to calculate the length of the third side. We can do this by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the
sum of the squares of the two shorter sides. If we denote the lengths of the two shorter sides as π and π and the length of the
hypotenuse as π, then this can be expressed as π squared plus π squared equals π
squared.

In our triangle, π·πΆ is the hypotenuse as it is directly opposite the right
angle. So we can form the equation π·πΆ squared equals nine squared plus 12 squared. To solve for π·πΆ, we evaluate the squares and find their sum, giving π·πΆ squared
equals 225. Finally, we take the square root of both sides of the equation, giving π·πΆ equals
15. Note that weβre only interested in the positive solution here as π·πΆ represents a
length.

We may notice that as all three side lengths of this triangle are integer values, it
is a Pythagorean triple. In fact, it is an enlargement of perhaps the most commonly known Pythagorean triple,
the right triangle with side lengths of three, four, and five units. Including the length units, weβve found that the length of line segment π·πΆ is 15
centimeters.