We are doing a project with Pythagorean Theorem where we need to measure a wall and then make pennants that fit on the string. I did that, my wall is 50in, my pennants were cut from a 9 by 9 square. When I measured the diagonal, it was 12.5, but when I used the Pythagorean Theorem it was 12.8. I am supposed to explain why they are different, but I don't know why they would be, especially since measuring it didn't come out right. 4 of them should have fit perfectly, and they were too long. Why didn't the measurement and the Pythagorean Theorem work out differently?

Question

We are doing a project with Pythagorean Theorem where we need to measure a wall and then make pennants that fit on the string. I did that, my wall is 50in, my pennants were cut from a 9 by 9 square. When I measured the diagonal, it was 12.5, but when I used the Pythagorean Theorem it was 12.8. I am supposed to explain why they are different, but I don't know why they would be, especially since measuring it didn't come out right. 4 of them should have fit perfectly, and they were too long. Why didn't the measurement and the Pythagorean Theorem work out differently?

Answer 1

9^2 + 9^2

= 81+81 = 162
√162 = 12.7279... = appr 12.8

Don't expect your "measurement" to match the accurate mathematical answer.
I could be your drawing, it could be your ruler, it could be your estimation.
Since the old imperial system of measurement using inches is not very
suitable for decimals, this could be another problem.

Answer 2

To understand why the measured diagonal and the diagonal calculated using the Pythagorean Theorem are different, let's first review the Pythagorean Theorem.

The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it can be represented as:

c^2 = a^2 + b^2

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

In your project, you have a square pennant with side lengths of 9 inches. When you cut this square into a right-angled triangle, one side of the triangle has a length of 9 inches, and the other side also has a length of 9 inches.

Using the Pythagorean Theorem, we can calculate the length of the hypotenuse (diagonal) of this triangle:

c^2 = 9^2 + 9^2
c^2 = 81 + 81
c^2 = 162
c ≈ √162
c ≈ 12.73

So according to the Pythagorean Theorem, the diagonal should be approximately 12.73 inches.

However, when you measured the diagonal, you obtained a length of 12.5 inches. This discrepancy between the measured value and the calculated value can be attributed to a few factors:

1. Measurement precision: Measuring tools may not provide absolute precision, and there could be slight errors in the measurement process. Even small inaccuracies can lead to differences in the measured value.

2. Cutting precision: While cutting the pennants from the square, there may have been slight differences in the angles or the straightness of the cuts, which can affect the resulting diagonal length.

3. Material properties: The material used for the pennants may have some flexibility or stretch, which could also lead to variations in the measured diagonal length.

It is important to note that the Pythagorean Theorem provides an exact relationship for right-angled triangles, assuming accurate measurements and precise calculations. However, in real-world scenarios, there can be various sources of error that can cause differences between the calculated and measured values.

To address the issue of the pennants being too long, you might want to double-check your measurements and make sure that the cuts are as precise as possible. Additionally, consider using more accurate measurement tools, such as a ruler with finer increments, to improve the overall precision of your calculations.