Question

The perimeter of parallelogram $\text{ABCD}$ is equal to $10\text{cm}$. Find the length of diagonal $\text{BD}$ if you know that the perimeter of $△\text{ABD}$ is $8\text{cm}$.

Answer 1

Find the length of the diagonal of the parallelogram:

• The given parallelogram is $\text{ABCD}$. A parallelogram has opposite sides parallel and equal and the perimeter of the polygon is the sum of all sides of the polygon.
• The perimeter of parallelogram $\text{ABCD}$ is $10\text{cm}$ and the perimeter of parallelogram is $\text{ABCD}$$=\text{AB}+\text{BC}+\text{CD}+\text{DA}$.

A parallelogram has opposite sides parallel and equal $\text{AB}=\text{CD, BC}=\text{DA}$. So,

$\begin{array}{rcl}\text{Perimeter of ABCD}& =& \text{AB}+\text{BC}+\text{CD}+\text{DA}\\ & \Rightarrow & 10=2\text{AB}+2\text{AD}\\ & \Rightarrow & 10=2\left(\text{AB+AD}\right)\\ & \Rightarrow & 5=\text{AB+AD}\end{array}$.

The perimeter of $△\text{ABD}$ is $8\text{cm}$.

Replace $\text{AB+AD}$ with $5$:

$\begin{array}{rcl}\text{Perimeter of}△\text{ABD}& =& \text{AB}+\text{BD}+\text{AD}\\ & \Rightarrow & 8=5+\text{BD}\\ & \Rightarrow & 3=\text{BD}\end{array}$.

Therefore, the length of diagonal $\text{BD}$ is $3\text{cm}$.