Question

Find the length of AG

Answer 1

58.27 mm (2 d.p.)

Explanation:

Assuming the given prism is a rectangular prism, triangles ADC and ACG are right triangles.

`\boxed{\begin{minipage}{9 cm}\underline{Cos trigonometric ratio} \\\\$\sf \cos(\theta)=(A)/(H)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}`

AC is the hypotenuse of right triangle ADC.

Therefore, we can use the cos trigonometric ratio to create an expression for the length of AC:

`\implies \cos\left(36^(\circ)\right)=(AD)/(AC)`

`\implies \cos\left(36^(\circ)\right)=(42)/(AC)`

`\implies AC=(42)/(\cos\left(36^(\circ)\right))`

AC is the hypotenuse of right triangle ACG.

Therefore, we can use the cos trigonometric ratio to create an expression for the length of AG:

`\implies \cos\left(27^(\circ)\right)=(AC)/(AG)`

`\implies AG=(AC)/(\cos\left(27^(\circ)\right))`

To find the length of AG, substitute the found expression for AC into the expression for AG:

`\implies AG=((42)/(\cos\left(36^(\circ)\right)))/(\cos\left(27^(\circ)\right))`

`\implies AG=(42)/(\cos\left(36^(\circ)\right)) * (1)/(\cos\left(27^(\circ)\right))`

`\implies AG=(42)/(\cos\left(36^(\circ)\right)\cos\left(27^(\circ)\right))`

`\implies AG=58.2654039...`

`\implies AG=58.27\;\sf mm \;(2\;d.p.)`