Chapter 5.1, Problem 12WE

To determine

To find: $ST$ and $SP$

Answer to Problem 12WE

  $SP=13\text{ cm}$

  $ST=14cm$

Explanation of Solution

Given:

  $STOP$ is a parallelogram.

The perimeter of $▱STOP$ is 54 cm.

The sideof the parallelogram $\overline{ST}$ is I cm longer then the side $\overline{SP}$ .

Concept Used:

Theorem 5-1: Opposite sides of a parallelogram are congruent.

Let the figure of the given parallelogram be:

Now, it is known that the perimeter of any polygon is given by the addition of the lengths of its sides.

So, the perimeter of $▱STOP$ is,

$\text{Perimeter of }▱STOP=ST+TO+PO+SP$ .... (1)

Now it is given that $▱STOP$ is a parallelogram.

So, by theorem 5-1, its opposite sides are congruent.

Here $\overline{ST}$ , $\overline{PO}$ and $\overline{SP}$ , $\overline{TO}$ are the opposite sides of the parallelogram $▱STOP$ . So,

$\overline{ST}\cong \overline{PO}$ And $\overline{SP}\cong \overline{TO}$

Thus,

  $ST=PO$

And

  $SP=TO$

So, equation (1), can be written as:

  $\begin{array}{c}\text{Perimeter of }▱STOP=ST+TO+PO+SP\\ =ST+SP+ST+SP\\ =2ST+2SP\\ =2\left(ST+SP\right)\end{array}$

Now it is given that the side $\overline{ST}$ of the parallelogram $▱STOP$ is I cm longer then the side $\overline{SP}$ .

So, let $SP=x\text{ cm}$ , then

  $ST=\left(x+1\right)\text{ cm}$

Substitute, $SP=x\text{ cm}$ , $ST=\left(x+1\right)\text{ cm}$ and the value of perimeter 54 cm in the above formula of perimeter.

  $\begin{array}{c}\text{Perimeter of }▱STOP=2\left(ST+SP\right)\\ 54=2\left(x+1+x\right)\\ 54=2\left(2x+1\right)\\ 54=4x+2\end{array}$

  $\begin{array}{c}54-2=4x\\ 52=4x\\ \frac{52}{4}=x\\ 13=x\end{array}$

Thus,

  $SP=13\text{ cm}$

And

  $\begin{array}{c}ST=\left(x+1\right)\text{ cm}\\ =\left(13+1\right)\text{ cm}\\ =14cm\end{array}$