Exercise 4 Page 783

To find the perimeter we need to add the four side lengths. However, in our parallelogram we are missing two of them. To find them we will consider the right triangle formed by the height, one of the sides, and a part of a nonparallel side.

In this triangle the measures of two of the interior angles are 90^(∘) and 45^(∘). We can use the Triangle Angle Sum Theorem to find the measure of the third angle. 180- 90- 45=45^(∘) The measure of the third angle of the triangle is 45^(∘). Therefore, we have a 45^(∘)-45^(∘)-90^(∘) triangle. In this type of special triangle, the length of the hypotenuse is sqrt(2) times the length of a leg. Here, the length of a leg is 5m. Hypotenuse: sqrt(2) * 5= 5sqrt(2)m The hypotenuse of this triangle is the a side of the parallelogram. Also, recall that opposite sides are congruent in a parallelogram. Therefore, the length of the opposite side is also 5sqrt(2)m.

Area

The area of a parallelogram is the product of a base and its corresponding height. We can consider the side whose length is 23m as the base. However, we need to find the height. To do so, we will pay close attention again to the right triangle formed on the right-hand side of the figure.

As previously mentioned, this is a 45^(∘)-45^(∘)-90^(∘) triangle. Since two of the angles are congruent, the triangle is an isosceles triangle and has two congruent sides. Therefore, the height is 5m.