Claire purchased just enough fencing to border either a rectangular or triangular garden as show, who perimeters are the same.
To determine the possible dimensions of the rectangular and triangular gardens, we need to consider the concept of perimeter. The perimeter of a shape is the total length of its boundary or the sum of the lengths of its sides.
Let's say the rectangular garden has length 'l' and width 'w', while the triangular garden has sides of lengths 'a', 'b', and 'c'. We can create equations to represent their perimeters:
Perimeter of the rectangular garden: P_rectangular = 2l + 2w
Perimeter of the triangular garden: P_triangular = a + b + c
Since Claire purchased just enough fencing to border either garden, the perimeters of both gardens must be the same. We can set up an equation to represent this:
2l + 2w = a + b + c
Now, we need to consider the number of variables and equations we have. We have three variables (l, w, and a, b, c) and only one equation. This means we can only solve for one variable in terms of the other variables.
Let's assume one variable is given or fixed. For example, let's assume l (length) is given. This allows us to express w and the sides of the triangle in terms of l:
w = (a + b + c - 2l) / 2
a = 2l - b - c
Now, we have expressed variables w and a in terms of l. We still have one variable (b or c) that we cannot eliminate, so let's assume another variable, such as b, is given:
c = 2l - a - b
With these expressions, we can find the possible dimensions of the rectangular and triangular gardens by assigning values to the given variables (l, w, a, b, c) and solving the equations accordingly.