Question

Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
1. 6:40
2. 5.57
3. 41
4. 9

Answer 1

To apply the Pythagorean Theorem, we need to treat the numbers provided as coordinates in a 2D space. Generally, the theorem is expressed as:

\[
c = \sqrt{a^2 + b^2}
\]

Where \(c\) is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the other two sides of a right triangle formed by the two points.

To proceed, let's assume that the numbers you provided represent the coordinates of two points as follows:

1. Point A: (6.40, 5.57)
2. Point B: (41, 9)

Now, we will calculate the length between these two points \(A\) and \(B\). We can find the differences in the coordinates:

\[
\Delta x = x_2 - x_1 = 41 - 6.40 = 34.60
\]
\[
\Delta y = y_2 - y_1 = 9 - 5.57 = 3.43
\]

Now we can use these differences in the Pythagorean Theorem:

\[
c = \sqrt{(34.60)^2 + (3.43)^2}
\]

Calculating the squares:

\[
(34.60)^2 = 1197.16
\]
\[
(3.43)^2 = 11.7649
\]

Adding these squares:

\[
1197.16 + 11.7649 = 1208.9249
\]

Now, we take the square root:

\[
c = \sqrt{1208.9249} \approx 34.77
\]

Rounding to the nearest hundredth, the length between the two points is approximately:

\[
\boxed{34.77}
\]