a landscape architect wants to make a triangle garden insde a square of land as shown at the right. What is a rule for the area A of the garden as a function of a?

To determine the rule for the area A of the garden as a function of a, let's analyze the given diagram. We have a square piece of land, and inside it, a triangle garden is to be created.

The side length of the square is labeled as "a". To determine the area of the garden, we need to calculate the area of the triangle. The triangle is formed by two sides of the square and the diagonal of the square.

To find the area of the triangle, we can use the formula for the area of a triangle: A = 1/2 * base * height. In this case, the base of the triangle is one side of the square, which has a length of "a". The height is the distance from the opposite vertex of the triangle to the base.

Since the opposite vertex is a right angle, we can consider the diagonal of the square as the height of the triangle. By using the Pythagorean theorem, we can determine the length of the diagonal. The Pythagorean theorem states that for a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (the hypotenuse).

In this case, the two shorter sides of the right triangle are each one side of the square, which is "a". Therefore, the length of the diagonal (which corresponds to the height of the triangle) can be calculated as √(a^2 + a^2) or √2a^2.

Now we have all the components to calculate the area of the triangle as A = 1/2 * a * √2a^2. Simplifying this expression, we get A = 1/2 * a * √2 * a.

Finally, the rule for the area A of the garden as a function of a is A(a) = 1/2 * a * √2 * a, which simplifies to A(a) = √2/2 * a^2.