Question

In figure HELP, is a parallelogram. If $OE = 4$cm and$HL$ is 5cm more than$PE$. Find the length of$OH$.

Answer 1

Hint: First use the property of the parallelogram that the diagonal of the parallelogram bisects each other to find the length$PO$ , which helps to find the length of the diagonal $PE$and then use the given condition which involves the value of the length of$PE$. Now, again use the property of the parallelogram to get the desired result that the problem is asking for.

Complete step-by-step answer:
It is given in the problem that $OE = 4$cm and$HL$ is 5cm more than$PE$. Then,
$OE = 4$cm and $HL = PE + 5$
The goal of the problem is to find the length of $OH.$
We know that the diagonals of a parallelogram bisect each other, which means that they divide each other into two equal parts. So, we have
$PO = OE = 4$cm
Now, we can find the length of the diagonal$PE$, we can express $PE$as
$PE = PO + OE$
Substitute the values $PO = OE = 4$cm so we have,
$PE = 4 + 4$
$PE = 8$cm
So, the length of the diagonal $PE$ is $8$cm.
We have given that $HL$ is 5cm more than$PE$, that is
$HL = PE + 5$
Substitute the value of $PE = 8$cm, and then we have
$HL = 8 + 5$
$HL = 8 + 5$
$HL = 13$cm
The length of the diagonal $HL$ is$13$cm.
As we know that the diagonals of the parallelogram bisect each other so we can write:
$OH = OL = \dfrac{1}{2}\left( {HL} \right)$
Substitute the value$HL = 13$cm, so we have
$OH = OL = \dfrac{1}{2}\left( {13} \right)$
$OH = OL = 6.5$cm
So, the required length $OH$ is$6.5$cm.

Note: Any parallelogram holds the following properties:
Opposite sides are parallel to each other;
Diagonals bisect each other;
Opposite sides have the same length.
We can use these properties in the given problem to get the desired result.