### Video Transcript

Find the area of a square whose
diagonal is nine centimeters.

Letโs begin by sketching this
square. Weโre told that its diagonal, which
is the straight line segment connecting opposite vertices of the square, is nine
centimeters long. We can then recall that the two
diagonals of the square are of equal length and that we can find the area of a
square with the diagonal of length ๐ units using the formula a half ๐ squared.

This is in fact the special case of
the formula for finding the area of a rhombus. The area of a rhombus with
diagonals of length ๐ one and ๐ two is a half ๐ one multiplied by ๐ two. A square is just a special type of
rhombus in which both diagonals are the same length. So replacing ๐ one and ๐ two in
this formula with ๐ gives the formula for the area of a square, using the length of
its diagonal.

Returning to our earlier formula
and substituting nine for the length of the squareโs diagonal gives that the area of
this square is a half multiplied by nine squared. Nine squared is 81, so this
simplifies to 81 over two. The units for area are square
units, so we have that the area of the square is 81 over two square centimeters.

An alternative approach, which we
wonโt go through in great detail here, would involve calculating the side length of
the square. We recall that each diagonal of a
square divides it into two congruent right triangles. Labeling the side length of the
square as ๐ centimeters, we could then use the Pythagorean Theorem to write down
the equation ๐ squared plus ๐ squared is equal to nine squared. This simplifies to two ๐ squared
equals 81, and then dividing both sides of the equation by two gives ๐ squared
equals 81 over two.

In fact, we can stop here and donโt
actually need to find the side length of the square at all. The area of a square is its side
length squared, which is exactly what we have here. This alternative method confirms
our answer, which is that the area of a square with a diagonal of nine centimeters
is 81 over two squared centimeters.