### Video Transcript

Calculate the area of triangle
π΄π΅π·.

We begin by recalling that the area
of a triangle is half the length of the triangleβs base multiplied by the triangleβs
perpendicular height. And since π΄π΅π· is a right
triangle at π·, we can choose π΅π· as the base and then π΄π· as the perpendicular
height. The area of the triangle is then
one over two times π΅π· multiplied by π΄π·. We can find the length π΄π· using
the Euclidean theorem and then the length π΅π· using the Pythagorean theorem.

If we first note that π· is the
projection of π΅ onto the side π΄πΆ and that the triangle π΄π΅πΆ is a right triangle
at π΅, then the Euclidean theorem tells us that π΄π΅ squared is equal to π΄πΆ
multiplied by π΄π·. Substituting now π΄π΅ is 44
centimeters and π΄πΆ is 55 centimeters, we have 44 squared is 55 multiplied by
π΄π·. And dividing through by 55, we have
44 squared over 55 is π΄π·, which evaluates to 35.2 centimeters. And we now have the lengths of two
sides of our right triangle π΄π΅π·. And so we can use the Pythagorean
theorem to find a third length, that is, the length π΅π·.

Applied to our triangle, the
Pythagorean theorem tells us that π΄π΅ squared is π΄π· squared plus π΅π·
squared. And substituting π΄π΅ is 44
centimeters and π΄π· is 35.2 centimeters, we have 44 squared is 35.2 squared plus
π΅π· squared. And subtracting 35.2 squared from
both sides and rearranging, this gives us π΅π· squared is 44 squared minus 35.2
squared. Our right-hand side evaluates to
696.96. And taking the positive square root
on both sides, since π΅π· is a length, thatβs the square root of 696.96, which is
equal to 26.4. So π΅π· is 26.4 centimeters.

So now, we can substitute π΄π· is
35.2 centimeters and π΅π· is 26.4 centimeters into our formula for the area. This gives us the area of triangle
π΄π΅π· is one over two times 26.4 multiplied by 35.2. This evaluates to 464.64, and hence
the area of triangle π΄π΅π· is 464.64 square centimeters.