What does 𝐴𝐶 squared equal
We’re asked what the square of the
length of side 𝐴𝐶 in the triangle shown is equal to. And to work this out, we’re going
to use areas. In fact, we’ll see that this is
actually one-half of the Euclidean theorem for right triangles.
We begin by redrawing triangle
𝐴𝐵𝐶, adding a square to each of its sides, and labeling the vertices of these
squares as shown. Our next step is to project 𝐴 onto
side 𝐺𝐻, meeting it at point 𝐾, and then adding the lines 𝐵𝐸 and 𝐴𝐺 to the
diagram as shown. We want to now show that triangles
𝐵𝐶𝐸 and 𝐺𝐶𝐴 are congruent. And from this, via equating some
areas, we’ll find an expression for 𝐴𝐶 squared.
The first thing we can note is that
angles 𝐵𝐶𝐸 and 𝐺𝐶𝐴 are congruent. This is because they’re both right
angles added to angle 𝐴𝐶𝐵. Now, we also know that side 𝐶𝐵 is
equal to side 𝐶𝐺 and that side 𝐴𝐶 is equal to side 𝐶𝐸. So we see that triangles 𝐵𝐶𝐸 and
𝐺𝐶𝐴 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
𝐺𝐶𝐴, where we choose 𝐺𝐶 as the base and where 𝐶𝐷 is the perpendicular height,
we have that the area of triangle 𝐺𝐶𝐴 is equal to one-half of 𝐺𝐶 times
𝐶𝐷. We know also that 𝐺𝐶 times 𝐶𝐷
is equal to the area of the rectangle 𝐶𝐷𝐾𝐺. So we have the area of triangle
𝐺𝐶𝐴 equals half the area of the rectangle 𝐶𝐷𝐾𝐺.
So, now making some space and
making a note of this, we can look similarly at triangle 𝐵𝐶𝐸. This time, we choose our base as
side 𝐸𝐶, and the perpendicular height is 𝐶𝐴. Then, the area of triangle 𝐵𝐶𝐸
equals one-half 𝐸𝐶 times 𝐶𝐴. Now we note that 𝐸𝐶 times 𝐶𝐴 is
the area of the square 𝐴𝐶𝐸𝐹. And we can observe also that since
triangles 𝐺𝐶𝐴 and 𝐵𝐶𝐸 are congruent, they must have the same area. This, in turn, means that the
rectangle 𝐶𝐷𝐾𝐺 must have the same area as the square 𝐴𝐶𝐸𝐹.
Now, the area of rectangle 𝐶𝐷𝐾𝐺
equals 𝐶𝐷 times 𝐷𝐾. And 𝐷𝐾 equals 𝐶𝐺, which is
equal to 𝐶𝐵. So the area of rectangle 𝐶𝐷𝐾𝐺
is equal to 𝐶𝐷 times 𝐶𝐵. Similarly, the area of the square
𝐴𝐶𝐸𝐹 equals 𝐴𝐶 times 𝐸𝐹. And since 𝐸𝐹 equals 𝐴𝐶, this
square’s area is equal to 𝐴𝐶 squared. So we have the area of the square
𝐴𝐶𝐸𝐹 equal to the area of the rectangle 𝐶𝐷𝐾𝐺. And so, 𝐴𝐶 squared is equal to
𝐶𝐷 multiplied by 𝐶𝐵, which is actually one part of the Euclidean theorem for