Matilde wants to build a wooden planter in the shape of a right triangle for their garden. One side will be formed by an existing bench that is 96 inches long. To fence the other two sides, Matilde wants to use pieces of wood that are each precut to 12 inches, so each of the two undetermined side lengths will need to be in whole number of inches. Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2 to find one of the other possible side lengths, assuming 2xy=96 and that the two undetermined sides are of different lengths. Which of the following values could be one of the side lengths for their wooden planter?(1 point)Responses20 inches20 inches80 inches80 inches100 inches100 inches24 inches

Answer 1

To solve this problem, we start with the polynomial identity given:

\[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \]

In this scenario, we are told that:

\[ 2xy = 96 \]

From this equation, we can solve for \(xy\):

\[ xy = \frac{96}{2} = 48 \]

Now, we can analyze our variables \(x\) and \(y\) in relation to a right triangle, where the two undetermined side lengths are \(a\) and \(b\), and the existing side length (which we denote as the hypotenuse for calculation purposes, although it's one of the sides of the triangle) is 96 inches.

Next, we know:

\[ a^2 + b^2 = c^2 \]

Let’s say for simplicity that \(x = a\) and \(y = b\). Then our identity becomes:

\[ (a^2 + b^2)^2 = (a^2 - b^2)^2 + (2ab)^2 \]

Using \(ab = 48\), we find \(2ab = 96\) fits our previous statement and we are valid in our use of this polynomial identity.

Now, we can derive further relationships. Since \(a\) and \(b\) are both integers whose product is 48 and are different in length, we enumerate the pairs of factors:

1. \(1 \times 48\)
2. \(2 \times 24\)
3. \(3 \times 16\)
4. \(4 \times 12\)
5. \(6 \times 8\)

Now let's check which of these can fit the Pythagorean theorem along with the existing side of 96 inches.

We can check pairs in terms of \(c\) (the existing bench's length) and the other two sides:

1. \(c^2 = 96^2 = 9216\)
2. We check each pair:

• \(1\) and \(48\): \(1^2 + 48^2 = 1 + 2304 = 2305\) (not a square)
• \(2\) and \(24\): \(2^2 + 24^2 = 4 + 576 = 580\) (not a square)
• \(3\) and \(16\): \(3^2 + 16^2 = 9 + 256 = 265\) (not a square)
• \(4\) and \(12\): \(4^2 + 12^2 = 16 + 144 = 160\) (not a square)
• \(6\) and \(8\): \(6^2 + 8^2 = 36 + 64 = 100\) (\(10^2\), is a square)

Thus, the pair \(6\) and \(8\) are valid side lengths that satisfy the relationship along with the existing side of \(96\). This means we can add the other two lengths to near \(96\):

• If we want the value from the list given in the options \(20\), \(80\), \(100\), and \(24\) we find:

The potential lengths \(6\) and \(8\) don't match but optimizing differently leads to subsequent evaluations based on whole numbers summing towards the triangle identification. Testing other factors finally lets you identify:

The strongest option to fit among \(20\), \(80\), \(100\), and \(24\) that can yield an integral component and using polynomial checks again on decomposed values referencing the earlier identities structuring would leave:

100 inches to become relevant to surface consideration hence align itself to \(c\).

Thus, the possibility for side lengths proves to be:

100 inches