Quilt squares are cut on the diagonal to form triangular quilt pieces. The hypotenuse of the resulting triangles is 16 inches long. What is the side length of each piece?
A) 8 in
B) 8√3 in
C) 16√2 in
D) 8√2 in
s^2 + s^2 = 16^2
2s^2 = 16^2
s = 16/√2
so, which choice is equal to that?
To find the side length of each triangular quilt piece, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the hypotenuse of the triangle is given as 16 inches. Let's call one side of the triangle "a" and the other side "b" (the hypotenuse being c). So, we have:
c^2 = a^2 + b^2
Since we are given the length of the hypotenuse as 16 inches, we can substitute this into the equation:
16^2 = a^2 + b^2
256 = a^2 + b^2
Now, we need to consider the fact that the quilt squares are cut on the diagonal, which forms a right triangle. The diagonal of a square divides it into two equal right triangles.
Since we know that the hypotenuse of the resulting triangles is 16 inches, we can find the length of one of the sides (a) by dividing the length of the hypotenuse by the square root of 2.
a = 16 / √2
To simplify this, we can multiply the numerator and denominator by the square root of 2:
a = 16 / √2 * √2 / √2
a = 16√2 / 2
a = 8√2 inches
Therefore, the side length of each triangular quilt piece is 8√2 inches.
The correct answer is (D) 8√2 inches.
To find the side length of each triangular quilt piece, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse of the triangle is 16 inches long. Let's label one of the other two sides as "a" and the remaining side as "b".
According to the Pythagorean Theorem, we have:
a^2 + b^2 = 16^2
Since we want to find the side length of each piece, we are interested in the length of a or b. Let's solve for a.
a^2 + b^2 = 16^2
We know that the length of the hypotenuse is 16, so we have:
a^2 + b^2 = 256
We are given that these are quilt pieces, so it makes sense to assume that the side lengths are integers (whole numbers). Let's check the answer choices to see which one gives us a^2 + b^2 = 256.
A) 8 in
Checking a = 8:
8^2 + b^2 = 64 + b^2 ≠ 256
B) 8√3 in
Checking a = 8√3:
(8√3)^2 + b^2 = (64 * 3) + b^2 = 192 + b^2 ≠ 256
C) 16√2 in
Checking a = 16√2:
(16√2)^2 + b^2 = (256 * 2) + b^2 = 512 + b^2 ≠ 256
D) 8√2 in
Checking a = 8√2:
(8√2)^2 + b^2 = (64 * 2) + b^2 = 128 + b^2 = 256 if b = 8
Since only option D), 8√2 in, satisfies the equation a^2 + b^2 = 256, the side length of each piece is 8√2 in. Therefore, the correct answer is D).