The shaded portion of the rectangular lot shown above…
In solving this problem we first must recognize that the flower bed is the right triangle with sides of y yards, x yards, and z yards. We are given that the area of the bed (which is the right triangle) is 24 square yards. Since we know that area of a triangle is ½ Base x Height, we can say:
24 = ½(xy)
48 = xy
We also know that x = y + 2, so substituting in y + 2 for x in the area equation we have:
48 = (y+2)y
48 = y^2 + 2y
y^2 + 2y – 48 = 0
(y + 8)(y – 6) = 0
y = -8 or y = 6
Since we cannot have a negative length, y = 6.
We can use the value for y to calculate the value of x.
x = y + 2
x = 6 + 2
x = 8
We can see that 6 and 8 represent two legs of the right triangle, and now we need to determine the length of z, which is the hypotenuse. Knowing that the length of one leg is 6 and the other leg is 8, we know that we have a 6-8-10 right triangle. Thus, the length of z is 10 yards.
If you didn’t recognize that 6, 8, and 10 are the sides and hypotenuse of a right triangle, you would have to use the Pythagorean to find the length of the hypotenuse: 6^2 + 8^2 = c^2 → 36 + 64 = c^2 → 100 = c^2. The positive square root of 100 is 10, and thus the value of z is 10.