# Question #12cb7

Jul 23, 2017

The length is $= 4.1 c m$

#### Explanation:

The length of the sides are

$A B = B C = C D = D A = 6 c m$

The angle

$\hat{A} = \hat{C} = {40}^{\circ}$

Therefore,

$\hat{B} = \hat{D} = {180}^{\circ} - {40}^{\circ} = {140}^{\circ}$

To calculate the longer diagonal, we apply the cosine rule to the triangle $\Delta A C D$

$A {C}^{2} = A {D}^{2} + D {C}^{2} - 2 \cdot A D \cdot D C \cdot \cos \hat{D}$

$A {C}^{2} = {6}^{2} + {6}^{2} - 2 \cdot 6 \cdot 6 \cdot \cos {140}^{\circ}$

$= 36 + 36 + 72 \cdot \left(- 0.766\right)$

$= 72 \left(1 - 0.766\right) = 16.84$

Therefore,

$A C = \sqrt{16.84} = 4.1 c m$