Rhombus ABCD has perimeter 148, and one of its diagonals has length 24. How long is the other diagonal?

Guest Aug 18, 2017

edited by
Guest
Aug 18, 2017

#1**+2 **

**Rhombus ABCD has perimeter 148, and one of its diagonals has length 24.**

**How long is the other diagonal?**

Let **e **one of its diagonal** **= 24

Let **f **the other diagonal = ?

Let **a** be the side of the rhombus. All sides are equal.

\(\mathbf{a= \ ? }\\ \begin{array}{|rcll|} \hline 4a &=& \text{perimeter} \\ a &=& \frac{ \text{perimeter} } {4} \\ a &=& \frac{ 148 } {4} \\ \mathbf{a} & \mathbf{=} & \mathbf{37} \\ \hline \end{array}\)

\(\mathbf{f= \ ? }\\ \begin{array}{|rcll|} \hline \mathbf{4a^2} & \mathbf{=} & \mathbf{e^2+f^2 } \\\\ 4\cdot 37^2 &=& 24^2+f^2 \\ 4\cdot 1369 &=& 576 + f^2 \\ 5476 &=& 576 + f^2 \quad & | \quad -576 \\ 5476-576 &=& f^2 \\ 4900 &=& f^2 \\ 70 &=& f \\\\ \mathbf{f} & \mathbf{=} & \mathbf{70} \\ \hline \end{array}\)

The other diagonal is **70**

heureka
Aug 18, 2017

#2**+1 **

Here's another method....

The sides are all equal, so one side = 148/ 4 = 37

And the diagonals meet at right angles......so 1/2 the length of the known diagonal = 1/2 * 24 = 12

And this half diagonal and one of the rhombus sides will form the leg and hypotenuse of a right triangle.....and the other half diagonal will form the other leg.....and its length = sqrt [ 37^2 - 12^2] = sqrt [ 1225 ] = 35

So......the length of the other diagonal = 2 * 35 = 70

CPhill
Aug 18, 2017