Shape and Space 10
3 marks · NormalExplain why the area of a triangle with sides of lengths and , and included angle , can be written as .
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The area of a triangle is × base × height. In any triangle, the height can be written as if base is and angle C is the included angle, so substituting gives .
Must show: Base × perpendicular height formula, Use of or similar reasoning, Substitute into area formula
Also accept: Area with explained via right triangle argument
Common mistakes: Saying area is always ; Not explaining source of height
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Marking points
M1 · 1 mark · States area = × base × height or equivalent.
M2 · 1 mark · Correctly shows/defines that height = (or equivalent trig step).
A1 · 1 mark · Obtains and explains as triangle area.
Worked solution
The area of any triangle is given by × base × height. Choosing side as the base, the height from the opposite vertex onto this base can be expressed as , because it is the component of side perpendicular to . Substituting into the area formula gives area = .