Exercise 3 Page 783

A parallelogram is a quadrilateral where both of the pairs of opposite sides are parallel and congruent. Any side can be called the base of the parallelogram. Its height is the perpendicular distance between any two parallel bases. For the given parallelogram, we will find its perimeter and area one at a time.

Perimeter

Consider the given parallelogram.

We can see that the lengths of two nonparallel sides are $20\text{ cm}$ and $12\text{ cm}.$ Since opposite sides are congruent, the lengths of the other two sides are also $20\text{ cm}$ and $12\text{ cm}.$

To find the perimeter, we add these four side lengths.

Area

The area of a parallelogram is the product of a base and its corresponding height. We can consider the side whose length is $12\text{ cm}$ as the base. However, we need to find the height. To do so, we will pay close attention to the right triangle formed by the height, a side, and a part of a nonparallel side. We can see that the measure of two of the interior angles of the triangle are $60^{\circ }$ and $90^{\circ }.$ We can use the Triangle Angle Sum Theorem to find the measure of the third angle. The third angle measures $30^{\circ }$ and, therefore, we have a $30^{\circ }\text{-}60^{\circ }\text{-}90^{\circ }$ triangle. In this type of triangle the length of the longer leg is $\frac{\sqrt{3}}{2}$ times the length of the hypotenuse. With this information, and knowing that the hypotenuse measures $20\text{ cm},$ we can find the length of the longer leg. Therefore, the height of the parallelogram is $10\sqrt{3}\text{ cm}.$ Now that we know that the base is $12\text{ cm}$ and that the height is $10\sqrt{3}\text{ cm},$ we can substitute these values in the formula for the area of a parallelogram. The area to the nearest tenth is $207.8{\text{ cm}}^2.$