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Using the properties of angles made when one line cuts a pair of parallel lines, it is proved, as illustrated on the left, that the angles of a triangle ABC add up to 180∘. By considering separate areas, illustrated on the right, it can be shown that the area of the triangle is half that of the rectangle shown, and so the area of a triangle is ‘half base times height’.

If now A, B and C denote the angles of the triangle, and a, b and c the lengths of the sides opposite them, the following results hold: (i) The area of the triangle equals ½ bc sin A.(ii) The sine rule: where R is the radius of the circumcircle.(iii) The cosine rule: a2=b2+c2−2bc cos A, or, in another form,(iv) The tangent rule:(v) Hero's formula: Let s=½(a+b+c). Then the area of the triangle equals

(i) The area of the triangle equals ½ bc sin A.

(ii) The sine rule: where R is the radius of the circumcircle.

(iii) The cosine rule: a2=b2+c2−2bc cos A, or, in another form,

(iv) The tangent rule:

(v) Hero's formula: Let s=½(a+b+c). Then the area of the triangle equals

Subjects: Mathematics.


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