am building a birdhouse. the bottom is 5" wide by 7" long. the sides are 7" long, and cut at a 45 degree angle. I need to find the dimensions of the front and back, which must sit at the diagonal.

To find the dimensions of the front and back panels of your birdhouse, which will sit diagonally, you can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the diagonal of the front and back panels will be the hypotenuse of a right triangle, with the bottom and side panels as the other two sides. Let's label the dimensions of the front and back panels as x and y.

Now, we can use the Pythagorean theorem to set up an equation:

x² + 5² = 7²
y² + 5² = 7²

Simplifying the equations, we get:

x² + 25 = 49
y² + 25 = 49

Subtracting 25 from both sides, we get:

x² = 24
y² = 24

Taking the square root of both sides gives us:

x = √24
y = √24

Simplifying the square root of 24:

x ≈ 4.899
y ≈ 4.899

Therefore, the dimensions of the front and back panels, which must sit diagonally, are approximately 4.899 inches wide by 7 inches long.