In a rhombus, each side has length 6. One of the angles of the rhombus is $120^\circ.$ Find the area of the rhombus.

Question

In a rhombus, each side has length 6. One of the angles of the rhombus is $120^\circ.$ Find the area of the rhombus.

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Eva 3 weeks 2021-09-08T02:00:53+00:00 2 Answers 0

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    0
    2021-09-08T02:02:39+00:00

    Step-by-step explanation:

    Let the rhombus be $ABCD$, where $\angle DAB = 120^\circ$. Then $\angle ABC = 180^\circ – \angle DAB = 180^\circ – 120^\circ = 60^\circ$.

    [asy] unitsize(1 cm); pair A, B, C, D; A = (0,1); B = (sqrt(3),0); C = (0,-1); D = (-sqrt(3),0); draw(A–B–C–D–cycle); draw(A–C); label(“$A$”, A, N); label(“$B$”, B, E); label(“$C$”, C, S); label(“$D$”, D, W); label(“$6$”, (A + D)/2, NW); [/asy]

    Since $AB = BC$, triangle $ABC$ is equilateral. By the same argument, triangle $ACD$ is also equilateral. Each triangle has area

    \[\frac{\sqrt{3}}{4} \cdot 6^2 = 9 \sqrt{3},\]so the area of the rhombus is $2 \cdot 9 \sqrt{3} = \boxed{18 \sqrt{3}}$.

    0
    2021-09-08T02:02:46+00:00

    Answer:

    31.18 units²

    Step-by-step explanation:

    The diagonals bisect the angles.

    ⇒ Angle at half the rhombus = 120 ÷ 2 = 60°

    Both sides of the triangle are 6 units:

    ⇒ Half the rhombus is also an isosceles triangle.

    ⇒ the other angle is also 60°.

    Both angles are 60°,

    ⇒ the third angle is 180 – 60 – 60 = 60°

    ⇒ it is an equilateral triangle.

    Area of an equilateral triangle = √3/4 (6)² = 15.59 units²

    The rhombus is made up of two of the triangles.

    ⇒ Area of the rhombus = 15.59 x 2 = 31.18 units²

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