Exercise 30 Page 784

For the given figure, we will find its perimeter and area one at a time.

Perimeter

Consider the given figure.

To find the perimeter we need to add the four side lengths. Notice that in our figure all the sides are congruent. This means that it is a square. To find the length of one side, we will consider the right triangle formed by the diagonal and two sides.

Notice that in this right triangle the legs are congruent. Therefore, we have a $45^{\circ }\text{-}45^{\circ }\text{-}90^{\circ }$ triangle. In this type of triangle, the length of each leg is $\frac{\sqrt{2}}{2}$ times the length of the hypotenuse. Since we know that the length of the hypotenuse is $2\sqrt{2}$ meters, we can find the length of the legs of this triangle. The legs of this triangle are the sides of our figure. All sides in our figure are congruent, so we found the side lengths of all the sides. Now we can add the four side lengths to obtain the perimeter.

Area

We know that the given figure is a square. The area of a square equals its side length squared. In this case the side length is $2\text{ m}.$ Let's find the area. The area of the square is $4$ square meters.