### Video Transcript

Find the length of the projection
of the line segment π΄πΈ on the line passing through π΅ and πΆ.

In this question, we are asked to
find the length of the projection of a line segment onto a line. To do this, letβs start by
recalling what is meant by the projection of a line segment onto a line. In general, this is the line
segment between the projection of the endpoints of the line segment. This is the same as saying that the
projection of the line segment between π and π on the line passing through π
and
π is the line segment between π prime and π prime, where π prime and π prime
lie on the line passing through π
and π. And the line segments ππ prime
and ππ prime are perpendicular with the line passing through π
and π.

To project the line segment between
π΄ and πΈ onto the line between π΅ and πΆ, letβs start by highlighting the two
objects on the diagram. Remember, the line passing through
π΅ and πΆ extends indefinitely in both directions. To project the line segment π΄πΈ
onto the line passing through π΅ and πΆ, we need to project points π΄ and πΈ onto
the line.

Letβs start by projecting point π΄
on the line. To project π΄ onto the line, we
want to find the point on the line such that π΄π΄ prime is perpendicular to the line
passing through π΅ and πΆ. We can see in the figure that the
line segment between π΄ and π΅ is perpendicular to the line passing through π΅ and
πΆ. So the projection of π΄ onto this
line is point π΅.

We can follow a similar process to
project point πΈ onto the line passing through π΅ and πΆ. We note that triangle πΆπΈπ· is an
isosceles triangle. And so we can recall that the
median from the base πΆπ· will be perpendicular to the base. So, πΈ prime lies at the midpoint
of line segment πΆπ·. Hence, the projection of the line
segment is the line segment between π΅ and πΈ prime. Remember, we want to find the
length of the projection. This means we want to find π΅πΈ
prime, which we can note is equal to π΅πΆ plus πΆπΈ prime. We can see in the diagram that
triangle π΄π΅πΆ is a right triangle. So we can find the length of line
segment π΅πΆ using the Pythagorean theorem. Its length is the square root of 58
squared minus 40 squared centimeters. We can evaluate this expression to
obtain 42 centimeters.

We can see in the diagram that line
segment πΆπ· has the same length as line segment π΅πΆ. And since πΈ prime is the midpoint
of line segment πΆπ·, the length of line segment πΆπΈ prime is half of 42. Hence, the length of the projection
of line segment π΄πΈ onto the line passing through π΅ and πΆ is 63 centimeters.