AREAS OF TRIANGLES Methods for Finding the Area of Oblique Triangles.

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AREAS OF TRIANGLES Methods for Finding the Area of Oblique Triangles

AREA FORMULA 1 – USING SINES The first Area Formula is based on our traditional formula: Area = ½ (Base)(Height) Area = ½ (Base)(Height) A C B c b a Given Triangle ABC, where we are given two sides and their included angle, the formula for the area of the triangle is derived as follows: Assume we are given sides b and c with angle A, then base height Area = ½ (c)(Height) b Sin A = h Area = ½ (c)(b Sin A) Similarly, we could have chosen sides a, b with angle C, Or we could have used sides a, c and angle B. For the highlighted right triangle,

EXAMPLE 1 In Triangle PMF, M=127 o, p=15.78, and f=8.54. Find the unknown measures of the triangle. Analyze – the information given is SAS, so we can use the Area Formula using Sines.

HERO’S FORMULA Our second area formula was derived by the ancient Greek mathematician Hero of Alexandria (ca. 100 BCE). Hero’s Formula For triangle ABC, the area can be found using Where a, b, and c are the side lengths of the triangle and s is the semi-perimeter (half of the perimeter)

EXAMPLE 2 In Triangle XYZ, x = 5cm, y = 8 cm, and z = 11 cm. Find the area of the triangle. 1. Find the semi-perimeter 2. Plug all values into the Hero’s formula.