# 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.

Please specify your process.

### 1 Answer

- micatkieLv 72 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