Answer: 7.5 square units
This value is exact and hasn't been rounded.
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Step-by-step explanation
Refer to the diagram shown below. I have plotted triangle STU inside of square ATBC. I used GeoGebra to make the diagram.
Square ATBC has side length 4, and area 4^2 = 16 square units. This will come in handy later.
Triangles TUA, TBS, and SCU have areas of 2, 2, and 4.5 square units in that order. Use the triangle area formula of area = 0.5*base*height.
Add those areas up to get 2+2+4.5 = 8.5 square units. This represents the unshaded white regions inside square ATBC.
Therefore the shaded region STU must have area 16-8.5 = 7.5 square units
Some alternative methods are:
- Use the polygon shoelace area formula. This is also called Gauss's area formula and another name for it is surveyor's formula.
- Use Heron's formula (first use the distance formula to compute the side lengths). This method may lead to rounding error.
- Use Picks Theorem