### Video Transcript

What does π΄πΆ squared equal
to?

Weβre asked what the square of the
length of side π΄πΆ in the triangle shown is equal to. And to work this out, weβre going
to use areas. In fact, weβll see that this is
actually one-half of the Euclidean theorem for right triangles.

We begin by redrawing triangle
π΄π΅πΆ, adding a square to each of its sides, and labeling the vertices of these
squares as shown. Our next step is to project π΄ onto
side πΊπ», meeting it at point πΎ, and then adding the lines π΅πΈ and π΄πΊ to the
diagram as shown. We want to now show that triangles
π΅πΆπΈ and πΊπΆπ΄ are congruent. And from this, via equating some
areas, weβll find an expression for π΄πΆ squared.

The first thing we can note is that
angles π΅πΆπΈ and πΊπΆπ΄ are congruent. This is because theyβre both right
angles added to angle π΄πΆπ΅. Now, we also know that side πΆπ΅ is
equal to side πΆπΊ and that side π΄πΆ is equal to side πΆπΈ. So we see that triangles π΅πΆπΈ and
πΊπΆπ΄ are congruent since they have at least two congruent sides with the included
angle between those sides also congruent.

Now, we know that the area of any
triangle is half the base times the perpendicular height. So, now applying this to triangle
πΊπΆπ΄, where we choose πΊπΆ as the base and where πΆπ· is the perpendicular height,
we have that the area of triangle πΊπΆπ΄ is equal to one-half of πΊπΆ times
πΆπ·. We know also that πΊπΆ times πΆπ·
is equal to the area of the rectangle πΆπ·πΎπΊ. So we have the area of triangle
πΊπΆπ΄ equals half the area of the rectangle πΆπ·πΎπΊ.

So, now making some space and
making a note of this, we can look similarly at triangle π΅πΆπΈ. This time, we choose our base as
side πΈπΆ, and the perpendicular height is πΆπ΄. Then, the area of triangle π΅πΆπΈ
equals one-half πΈπΆ times πΆπ΄. Now we note that πΈπΆ times πΆπ΄ is
the area of the square π΄πΆπΈπΉ. And we can observe also that since
triangles πΊπΆπ΄ and π΅πΆπΈ are congruent, they must have the same area. This, in turn, means that the
rectangle πΆπ·πΎπΊ must have the same area as the square π΄πΆπΈπΉ.

Now, the area of rectangle πΆπ·πΎπΊ
equals πΆπ· times π·πΎ. And π·πΎ equals πΆπΊ, which is
equal to πΆπ΅. So the area of rectangle πΆπ·πΎπΊ
is equal to πΆπ· times πΆπ΅. Similarly, the area of the square
π΄πΆπΈπΉ equals π΄πΆ times πΈπΉ. And since πΈπΉ equals π΄πΆ, this
squareβs area is equal to π΄πΆ squared. So we have the area of the square
π΄πΆπΈπΉ equal to the area of the rectangle πΆπ·πΎπΊ. And so, π΄πΆ squared is equal to
πΆπ· multiplied by πΆπ΅, which is actually one part of the Euclidean theorem for
right triangles.