Anne is having a new front gate made and has decided on the design below - Leaving Cert Mathematics - Question (b) - 2012

To find the angle EGF, consider triangle ( \triangle AFB ):

• We know the opposite side is 2 metres (width) and the adjacent side is 1 metre (height from A to line FG).

Using the tangent function:

$\tan(\angle AFB) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{1}$

Thus:

$\angle AFB = \tan^{-1}(2) \approx 63.43^{\circ}$

Now, in triangle ( \triangle EGF ), since the angles inside triangle EGF must sum to 180°, we have:

$\angle EGF + \angle AFB + \angle ZEGF = 180°$

Hence:

$\angle EGF = 180° - 63.43° - 53° = 63.43°$

To the nearest degree, ( \angle EGF \approx 63 °$$.

Using the sine rule in triangle ( \triangle EGF ), we can find the length |GE|:

Given:

• We have already calculated |EG| to be 0.5 meters and ( \angle EGF \approx 63.43° ):

Using the sine rule:

$\frac{|GE|}{\sin(63.43)} = \frac{0.5}{\sin(53.14)}$

Calculating |GE|:

$|GE| = \frac{0.5 \cdot \sin(63.43)}{\sin(53.14)} \approx 0.558 \text{ metres, correct to three decimal places.}$

Thus, the common distance |AG| and |DC| is approximately 1.678 metres.