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Trigonometry

Question 7 of 9

Area of Triangle

Question

A triangular backyard of \(40\,\text{m}^2\) area with an angle measuring \(80^\circ\) opposite to the house is needed to be fenced. Having one side measuring \(10\,\text{m}\) already fenced, how many metres of fence are needed for the other side? (Round your answer to 2 decimal places)

Show solution Hide solution Fully worked — 4 steps
Triangle ABC, drawn to scale, with vertices A at the top left and B at the top right; the side between them, labelled HOUSE, is the side of the backyard that runs along the house. Vertex C is at the bottom, where the given angle of 80 degrees, opposite the house, is marked with an arc. The already-fenced side b = 10 runs from A down to C; the unknown side a, from B down to C, is the side the solution finds.
  1. State the formula
    \[ \mathit{area} = \frac{1}{2}\,ab \sin C \]
  2. Substitute values based on formula
    \[ 40\,\text{m}^2 = \frac{1}{2}(10\,\text{m})(a)(\sin 80^\circ) \]
  3. Solve for \(a\)

    Evaluate \(\dfrac{1}{2}(10\,\text{m})(a)(\sin 80^\circ)\):

    \[ 40 = 4.9240a \]

    Divide both sides by 4.9240 to get \(a\):

    \[ a = \frac{40}{4.9240} \]
  4. Final answer

    Divide 40 by 4.9240:

    \[ a = 8.12\,\text{m} \]

    That is, the fence needed for the other side is 8.12 metres, to 2 decimal places.