Question

Anja is creating a triangle using three straws. The two shorter straws have lengths of 20 cm and 21 cm. How long, in centimeters, should the third straw be for Anja to create a right triangle?

Answer 1

To form a right triangle with straws (sides of the triangle), 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.

Let’s denote the lengths of the two shorter straws as \( a = 20 \) cm and \( b = 21 \) cm. The length of the third straw, which we will denote as \( c \), must satisfy the conditions of the Pythagorean theorem depending on whether it is shorter than or longer than the hypotenuse.

### Case 1: \( c \) is the hypotenuse
If the third straw is the longest side, then:
\[
c^2 = a^2 + b^2
\]
Calculating \( a^2 \) and \( b^2 \):
\[
a^2 = 20^2 = 400
\]
\[
b^2 = 21^2 = 441
\]
Adding these:
\[
c^2 = 400 + 441 = 841
\]
Taking the square root gives:
\[
c = \sqrt{841} = 29 \text{ cm}
\]

### Case 2: \( c \) is not the hypotenuse
If \( c \) is one of the shorter sides, we can determine if that can create a right triangle. In this case, we can determine that the hypotenuse must remain either 21 cm or 20 cm. Thus we check:

1. If \( c = 20 \):
\[
21^2 = 20^2 + c^2
\]
This gives us:
\[
441 = 400 + c^2 \Rightarrow c^2 = 441 - 400 = 41
\]
Therefore, \( c = \sqrt{41} \approx 6.4 \) cm.

2. If \( c = 21 \):
\[
20^2 + c^2 = 21^2
\]
Which does not yield a valid result.

Combining these results, we find that if Anja wants to use the three straws to create a right triangle, the valid lengths for the third straw \( c \) are \( 29 \) cm (as the hypotenuse) or \( \sqrt{41} \) cm (approximately \( 6.4 \) cm).

### Conclusion
Thus, the lengths of the third straw \( c \) that would allow Anja to create a right triangle are:

- **29 cm** (as the hypotenuse)
- **approximately 6.4 cm** (if it is one of the legs)

If we only seek the integer length for the third straw, the answer would be:
\[
\boxed{29} \text{ cm}
\]