Question

The two adjacent sides of a rectangle are $2{{x}^{2}}-5xy+3{{z}^{2}}$ and $4xy-{{x}^{2}}-{{z}^{2}}$. Find the perimeter of the rectangle.

Answer 1

Hint: For a rectangle the opposite pair of sides are equal. Hence we will have two sides whose length is equal to $2{{x}^{2}}-5xy+3{{z}^{2}}$ and two whose length is equal to $4xy-{{x}^{2}}-{{z}^{2}}$. Use the perimeter of a rectangle = Sum of lengths of all sides.

Complete step-by-step answer:

Monomial: An algebraic expression having only one term is called a monomial,
 e.g. ${{x}^{3}}{{y}^{2}}z$ etc.
Binomial: An algebraic expression having exactly two terms is called a binomial, e.g. $xy+yz$ etc.
Trinomial: An algebraic expression having exactly three terms is called a trinomial, e.g. $xy+yz+xz$ etc.
Polynomial: An algebraic expression having one or more than one terms is called a polynomial, e.g. \[{{a}^{3}}+{{b}^{3}}+3{{a}^{2}}b+3a{{b}^{2}}\] etc.
The letters a, b, c, etc. in algebraic expressions are called variables, and the numbers 1, 3, 1.5, etc. are called the constants.
When adding two algebraic expressions which are similar, e.g. term 7x will be added to 3x but not $3{{x}^{2}}$
Since a rectangle is a special case of a parallelogram and in a parallelogram opposite pair of sides are of equal length. Hence in the rectangle above two sides will have length $2{{x}^{2}}-5xy+3{{z}^{2}}$ , and two will have length $4xy-{{x}^{2}}-{{z}^{2}}$. Hence the perimeter of the rectangle will be
$P=4xy-{{x}^{2}}-{{z}^{2}}+4xy-{{x}^{2}}-{{z}^{2}}+2{{x}^{2}}-5xy+3{{z}^{2}}+2{{x}^{2}}-5xy+3{{z}^{2}}$
Combining similar terms together, we get
$\begin{align}
  & P=\left( 4xy+4xy-5xy-5xy \right)+\left( -{{x}^{2}}-{{x}^{2}}+2{{x}^{2}}+2{{x}^{2}} \right)+\left( -{{z}^{2}}-{{z}^{2}}+3{{z}^{2}}+3{{z}^{2}} \right) \\
 & =-2xy+2{{x}^{2}}+4{{z}^{2}} \\
\end{align}$
Hence the perimeter of the rectangle $=-2xy+2{{x}^{2}}+4{{z}^{2}}$

Note: Alternatively, we know p=2(l+b)
Here $l=2{{x}^{2}}-5xy+3{{z}^{2}}$ and $b=4xy-{{x}^{2}}-{{z}^{2}}$
Hence $p=2\left( l+b \right)=2\left( 2{{x}^{2}}-5xy+3{{z}^{2}}+4xy-{{x}^{2}}-{{z}^{2}} \right)$
Combining similar terms together, we get
$\begin{align}
  & p=2\left( \left( 4xy-5xy \right)+\left( 2{{x}^{2}}-{{x}^{2}} \right)+\left( 3{{z}^{2}}-{{z}^{2}} \right) \right) \\
 & =2\left( -xy+{{x}^{2}}+2{{z}^{2}} \right) \\
 & =-2xy+2{{x}^{2}}+4{{z}^{2}} \\
\end{align}$
which is the same as obtained above.