A triangle has sides of length 10 cm, 8 cm and 9 cm.(a) Calculate, in degrees to the nearest 0.1∘ , the size of the largest angle of this triangle.(b) Find, to 3 significant figures, the area of this triangle.
Question
A triangle has sides of length 10 cm, 8 cm and 9 cm.(a) Calculate, in degrees to the nearest 0.1∘ , the size of the largest angle of this triangle.(b) Find, to 3 significant figures, the area of this triangle.
Solution
(a) To find the largest angle in a triangle, we can use the Law of Cosines. The formula is:
cos(C) = (a² + b² - c²) / 2ab
where:
- C is the angle you are trying to find
- a and b are the lengths of the sides forming angle C
- c is the side opposite angle C
In this case, the largest angle would be opposite the longest side, which is 10 cm. So, a = 8 cm, b = 9 cm, and c = 10 cm. Substituting these values into the formula, we get:
cos(C) = (8² + 9² - 10²) / 289 cos(C) = (64 + 81 - 100) / 144 cos(C) = 45 / 144 cos(C) = 0.3125
To find the angle, we take the inverse cosine (cos⁻¹) of 0.3125. Using a calculator, we find that:
C = cos⁻¹(0.3125) = 71.6° (to the nearest 0.1°)
(b) To find the area of the triangle, we can use Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
where:
- s is the semi-perimeter of the triangle, calculated as (a + b + c) / 2
- a, b, and c are the lengths of the sides of the triangle
In this case, a = 8 cm, b = 9 cm, and c = 10 cm. So, s = (8 + 9 + 10) / 2 = 13.5 cm. Substituting these values into the formula, we get:
Area = √[13.5(13.5 - 8)(13.5 - 9)(13.5 - 10)] Area = √[13.5 * 5.5 * 4.5 * 3.5] Area = √[1215.15625] Area = 34.841 cm² (to 3 significant figures)
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