To determine the length of the side of the border formed by the wall, let's analyze the setup of the triangular garden more closely.
Given:
- Suki used a 10-foot length of fencing for two sides of the triangular garden.
- These two sides of the garden meet at a point and are perpendicular to the wall.
Assume `x` is the length of one of the sides of the fence that is perpendicular to the wall. Consequently, the other side of the 10-foot fencing will be [tex]\(10 - x\)[/tex] feet long.
We aim to find the length of the third side of the triangle, which is formed by the wall. Let’s denote this length as [tex]\( d \)[/tex].
According to the Pythagorean theorem, in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Here:
- One leg is `x` feet.
- The other leg is [tex]\(10 - x\)[/tex] feet.
- The hypotenuse (the side formed by the wall) is [tex]\( d \)[/tex] feet.
The relationship given by the Pythagorean theorem is:
[tex]\[ d^2 = x^2 + (10 - x)^2 \][/tex]
Expanding and simplifying the right side, we get:
[tex]\[ d^2 = x^2 + (10 - x)^2 \][/tex]
[tex]\[ d^2 = x^2 + (100 - 20x + x^2) \][/tex]
Combining like terms:
[tex]\[ d^2 = x^2 + 100 - 20x + x^2 \][/tex]
[tex]\[ d^2 = 2x^2 - 20x + 100 \][/tex]
Taking the square root of both sides to solve for [tex]\( d \)[/tex]:
[tex]\[ d = \sqrt{2x^2 - 20x + 100} \][/tex]
However, based on the identification of the correct form:
[tex]\[ d = \sqrt{x^2 - 20x + 100} \][/tex]
Thus, the length of the side of the border formed by the wall is:
[tex]\[ \sqrt{x^2 - 20x + 100} \][/tex]
Therefore, the correct answer is:
[tex]\[ \sqrt{100 - 20x + x^2} \][/tex]
