Question

In a parallelogram, $OS = 5cm$ and $PR$ is $6cm$ more than $QS$. Find $OP$.

Answer 1

Hint: To solve this question you have to use the property which states that diagonals of parallelogram bisect each other equally and then put the given values in equations and find the answer.

Complete step by step answer:

Given: $PQRS$ is a parallelogram
$OS = 5cm$
Using the property of parallelogram which states that diagonals of the parallelogram bisect each other equally which means the intersection point of both the diagonals is the midpoint of the diagonals.
So O point will be the mid point of line SQ and PR.
$OS = OQ$--(1) and $PO = OR$ ---(2)
We know
$SQ$ is made from 2 parts which are $OS$ and $OQ$ or is the midpoint of side $SQ$.
$SQ = OS + OQ$
Putting the value of $OQ$ using equation (1)
$
  SQ = 2OS \\
  OS = 5 cm \\
    \\
 $
So, $SQ = 10cm$--(3)
And we are given that $PR$ is $6cm$ more than $QS$. This means
$PR = QS + 6$
Putting the value of side $QS$ using equation (3)
$PR = 10 + 6$
$PR = 16cm$---(4)
So ,length of the side PR will be 16cm
And we know O is the midpoint of $PR$, so we can make an equation
$PR = OP + OR$
Putting the value of side OP using equation (2)
So ,side $PR$ will be the twice of side $OP$
$
  PR = 2OP \\
  OP = \dfrac{1}{2}PR \\
 $
Now put the value of side PR using equation (4)
$OP = \dfrac{1}{2} \times 16$
$OP = 8cm$
Therefore , the length of side $OP$ is $8cm$


Note:
A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle.