### Video Transcript

Given that π΄π΅ equals 29, πΆπ΅
equals 20, and πΆπ· equals 35, calculate the length of the projection of line
segment πΆπ· on line π΄π·.

Before we can calculate the length
of this projection, weβll need to see which line segment this projection is. Line π΄π· is our target line. Thatβs the line upon which the
projection will fall. And πΆπ· is the line segment that
weβll use to create our projection. But in order to create a
projection, weβll need perpendicular lines to our target line. Because we know that line π΅πΆ and
line π΄π· are parallel, the angle π΅πΆπ΄ is an alternate interior angle to the angle
πΆπ΄π·, which means the line segment π΄πΆ is perpendicular to the line segment
π΄π·.

Once we imagine the light source as
the set of all the perpendicular lines to our target line, we can find the endpoints
of our projection. Starting at πΆ, the projection of
point πΆ onto π΄π· would be the point π΄ and the projection of point π· onto π΄π·
would be itself. And that means the projection of
πΆπ· is going to be equal to π΄π·. This is the value we want to
calculate the length of. To do that, letβs clear out our
light source lines and think about what we know about right triangles.

First of all, we know that π΄π΅ has
a measure of 29 and πΆπ΅ has a measure of 20 and πΆπ· has a measure of 35. What we have in our figure is two
separate right triangles that make up this quadrilateral. And so we remember that we can find
side lengths in right triangles using the Pythagorean theorem, where π and π
represent the two smaller sides and π represents the hypotenuse. π squared plus π squared equals
π squared. To use the Pythagorean theorem, you
need at least two of the lengths. Since we donβt know the length of
π΄πΆ, we canβt find the length of π΄π·. However, we can use the information
about the smaller triangle π΄π΅πΆ to solve for the side length π΄πΆ first.

When we plug in what we know, we
get 20 squared plus π΄πΆ squared equals 29 squared. 400 plus π΄πΆ squared equals
841. So weβll subtract 400 from both
sides of the equation, and weβll get π΄πΆ squared equals 441. After that, weβll take the square
root of both sides. Weβre only interested in the
positive square root since weβre dealing with distance, and so we see that π΄πΆ
equals 21. Now that we know that π΄πΆ is 21,
we know two distances in our right triangle and weβll be able to find the side
length of our third distance π΄π·. But weβll need to set up the
Pythagorean theorem for a second time.

This time, weβll have 21 squared
plus π΄π· squared equals 35 squared. 441 plus 80 squared equals
1225. And so we subtract 441 from both
sides, and weβll get π΄π· squared equals 784. So we take the square root of both
sides. Again, weβre only interested in the
positive square root of 784, which is 28. The line segment π΄π· will be equal
to 28. The line segment π΄π· is the
projection of the line segment πΆπ· onto line π΄π·, and it has a measure of 28.