Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

Show that area I + area Il area Ill. ?

An equilateral triangle is inscribed in, and has a 

common vertex with, a square. 

Show that area I + area Il area Ill. 

Update:

Please specify your process.

Attachment image

1 Answer

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  • 2 months ago

    Refer to figure 1 below.

    The diagonal of the square bisects one of the interior angle.

    θ₁ = θ₂ = 45° - 30° = 15°

    Let a be the length of each side of the equilateral triangle.

    Let b be the length of each arm of the right triangle III.

    In right triangle III:

    a² = b² + b²  (Pythagorean theorem)

    b² = a²/2

    Area III

    = (1/2) b²

    = (1/2) a²/2

    = a²/4 …… [1]

    Refer to Figure below.

    Move triangle I to the top of triangle II as shown. The two triangles combine to form a isosceles triangle which is shaded.

    Area I + Area II

    = Area of the shaded triangle

    = (1/2) a² sin(θ₁ + θ₂)

    = (1/2) a² sin(15° + 15°)

    = (1/2) a² (1/2)

    = a²/4 …… [2]

    [2] = [1]:

    Hence, Area I + Area II = Area III

    Attachment image
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