Imagine a rather large slice of pizza in the shape of the sector of a circle roughly the same as the slice shown in the photograph above. If the slice can be cut exactly in half with a straight line from A to B, what is the angle at the point of the sector?

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Let the angle at the point of the sector be \(a\) degrees. Let the radius of the whole pizza circle be \(r\) cm.

The area of the triangle made by the cut is \( \frac12 r^2 \sin a \)

The area of the whole sector is \( \frac{a}{360} \pi r^2 \)

The segment made by the cut has an area equal to the area of the sector minus the area of the triangle so the area of the sector must be twice the area of the triangle.

$$ \frac{a}{360} \pi r^2 = 2 \times \frac12 r^2 \sin a$$

$$ a \pi = 360 \times \sin a $$

Using the GDC to solve this equation gove the angle as 109^{o} correct to three significant figures.

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