Answer :
Answer:
The side length is 9 cm and the diagonal's length is [tex]9\sqrt2[/tex] cm.
Step-by-step explanation:
Features of a Square
Area
A square is a special quadrilateral that not only have equal side lengths but, 90 degree angles in all its vertices!
Because of this, when finding its area or multiplying its "length" by its "width" the square's area formula looks like [tex]A=s^2[/tex], where s is the side length.
Diagonals
The diagonal of a quadrilateral "starts" from one corner or vertice of the shape and "ends" on the opposite corner. For either diagonal on a square, a diagonal bisects the vertices' angle measure or splits it into 45 degrees and 45 degrees.
The diagonal creates two 45-45-90 special right triangles.
The legs of the right triangles or, the side lengths of the square, are the same length, aligning to the proportion of a 45-45-90 triangle:
[tex]\boxed{x:x}:x\sqrt2[/tex],
thus proving the statement above!
Applying the Features of Square
If we're given A then we can plug it into the area equation to get the side length of a square!
[tex]81=s^2\\\\\sqrt{81} =s^2\\\\s=\pm9[/tex]
Recall that no geometric shape can have a negative side length so,
the side length of the square is +9 cm!
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If the diagonal of the square, that we know has a length of 9 cm, is also the hypotenuse of the right triangle it creates, then we apply the proportion of a 45-45-90 triangle,
[tex]x:x:x\sqrt2[/tex],
[tex]\Longrightarrow 9:9:9\sqrt2[/tex].
The diagonal's length is [tex]9\sqrt2[/tex] cm!
