Video Transcript
Find the length of the projection
of the line segment 𝐴𝐸 on the line passing through 𝐵 and 𝐶.
In this question, we are asked to
find the length of the projection of a line segment onto a line. To do this, let’s 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 𝑃 and 𝑄 on the line passing through 𝑅 and
𝑆 is the line segment between 𝑃 prime and 𝑄 prime, where 𝑃 prime and 𝑄 prime
lie on the line passing through 𝑅 and 𝑆. And the line segments 𝑃𝑃 prime
and 𝑄𝑄 prime are perpendicular with the line passing through 𝑅 and 𝑆.
To project the line segment between
𝐴 and 𝐸 onto the line between 𝐵 and 𝐶, let’s start by highlighting the two
objects on the diagram. Remember, the line passing through
𝐵 and 𝐶 extends indefinitely in both directions. To project the line segment 𝐴𝐸
onto the line passing through 𝐵 and 𝐶, we need to project points 𝐴 and 𝐸 onto
the line.
Let’s start by projecting point 𝐴
on the line. To project 𝐴 onto the line, we
want to find the point on the line such that 𝐴𝐴 prime is perpendicular to the line
passing through 𝐵 and 𝐶. We can see in the figure that the
line segment between 𝐴 and 𝐵 is perpendicular to the line passing through 𝐵 and
𝐶. So the projection of 𝐴 onto this
line is point 𝐵.
We can follow a similar process to
project point 𝐸 onto the line passing through 𝐵 and 𝐶. We note that triangle 𝐶𝐸𝐷 is an
isosceles triangle. And so we can recall that the
median from the base 𝐶𝐷 will be perpendicular to the base. So, 𝐸 prime lies at the midpoint
of line segment 𝐶𝐷. Hence, the projection of the line
segment is the line segment between 𝐵 and 𝐸 prime. Remember, we want to find the
length of the projection. This means we want to find 𝐵𝐸
prime, which we can note is equal to 𝐵𝐶 plus 𝐶𝐸 prime. We can see in the diagram that
triangle 𝐴𝐵𝐶 is a right triangle. So we can find the length of line
segment 𝐵𝐶 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 𝐶𝐷 has the same length as line segment 𝐵𝐶. And since 𝐸 prime is the midpoint
of line segment 𝐶𝐷, the length of line segment 𝐶𝐸 prime is half of 42. Hence, the length of the projection
of line segment 𝐴𝐸 onto the line passing through 𝐵 and 𝐶 is 63 centimeters.
