Find the diagonal

Geometry Level 2

The perimeter of a rhombus of 96 cm and obtuse angle of 12 0 ∘ 120^\circ .

Find the lengths of its diagonals.

24cm & 41.57cm 25cm & 42.87cm 21cm & 43.67cm 23cm & 40.56cm

Ojasee Duble by Ojasee Duble , 19, India

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2 solutions

Saya Suka
Dec 21, 2016

No other calculation apart from dividing 96 into 4 equal side length needed. If the obtuse angle is 120°, then said rhombus must have made up from combining 2 equilateral triangle together, so its shorter diagonal must be 24.

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Consider the diagram on the left.

By law of cosines, we have

a 2 = 2 4 2 + 2 4 2 − 2 ( 24 ) ( 24 ) ( cos ⁡ 120 ) a^2=24^2+24^2-2(24)(24)(\cos~120) ⟹ \implies a = 41.57 a=41.57

By law of cosines again, we have

b 2 = 2 4 2 + 2 4 2 − 2 ( 24 ) ( 24 ) ( cos ⁡ 60 ) b^2=24^2+24^2-2(24)(24)(\cos~60) ⟹ \implies b = 24 b=24

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