Question

$ABCD$ is a parallelogram, $AB=14cm,BC=18cm$ and $AC=15cm$. Find the length of the other diagonal.

• A. $24cm$
• B. $35cm$
• C. $28cm$
• D. None of These
  

Answer 1

The correct option is D None of These

Referring to the figure,

Let,

$AB=CD=a=14$

$BC=AD=b=18$

$AC=d_1=15$

$BD=d_2$

$AE=DF=x$

$BE=CF=k$

Applying Pythagorus theorem in $△CFA$, we get

$\Rightarrow k^2+(b-x{)}^2=d_1^2$

$\Rightarrow k^2+(18-x{)}^2={15}^2$...................$I$

Applying Pythagorus theorem in $△BDE$, we get

$\Rightarrow k^2+(b+x{)}^2=d_2^2$

$\Rightarrow k^2+(18+x{)}^2=d_2^2$...................$II$

Adding equations $I$ and $II$, we get

$\Rightarrow 2\times k^2+2\times {18}^2+2\times x^2={15}^2+d_2^2$

$\Rightarrow 2\times (k^2+{18}^2+x^2)={15}^2+d_2^2$..............$III$

Now applying the Pythagoras Theorem in $△BAE$, we get

$\Rightarrow k^2+x^2=a^2$

$\Rightarrow k^2+x^2={14}^2$...................$IV$

Putting equation $IV$ in $III$, we get

$\Rightarrow 2\times ({14}^2+{18}^2)={15}^2+d_2^2$

$\Rightarrow d_2=22.548$

Hence answer is option $D$