The side length of a square is 𝑥
centimeters, and the dimensions of a rectangle
are 𝑥 centimeters and two centimeters. Given that the sum of their areas
is eight square centimeters, determine the perimeter of the square.
If we consider the two diagrams,
the square with length 𝑥 centimeters and the rectangle with dimensions 𝑥
centimeters and two centimeters, we can calculate their areas by multiplying the
length by the width. The area of the square is 𝑥
squared, as 𝑥 multiplied by 𝑥 is 𝑥 squared. The area of the rectangle is two
𝑥, as two multiplied by 𝑥 is two 𝑥. We were told in the question that
the sum of their areas is eight square centimeters. This means that 𝑥 squared plus two
𝑥 is equal to eight. Subtracting eight from both sides
of the equation gives us 𝑥 squared plus two 𝑥 minus eight equals zero.
As this quadratic equation is equal
to zero, we can factorize it into two parentheses or brackets. Factorizing 𝑥 squared plus two 𝑥
minus eight gives us 𝑥 plus four multiplied by 𝑥 minus two, as four multiplied by
negative two is negative eight. And four plus negative two is
Solving this quadratic equation
gives us two possible values. Either 𝑥 plus four equals zero or
𝑥 minus two equals zero. We therefore have two values of
𝑥. Either 𝑥 equals negative four or
𝑥 equals positive two. As we are dealing with shapes, and
in this case squares and rectangles, our answer must be 𝑥 equals two, as 𝑥 must
have a positive value. A length must be positive.
As the length of the square is two
centimeters, the perimeter can be calculated by multiplying two by four. Two multiplied by four is equal to
eight. Therefore, the perimeter of the
square is eight centimeters. It is worth noting in this question
that the rectangle that was mentioned is also a square, as it too has dimensions two
centimeters by two centimeters.