A Pythagorean triple consists of three positive integers that work with the Pythagorean Theorem.

Question

Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 20 and 99.




Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 180 and 181.

Answer 1

To find the missing value in a Pythagorean triple where the lengths of the two known sides are given, we can use the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

1. For the right triangle with lengths 20 and 99:

Assuming that 99 is the hypotenuse and we are looking for the missing side \( a \):

\[ 20^2 + a^2 = 99^2 \]

Calculating the squares:

\[ 400 + a^2 = 9801 \]

Now, solve for \( a^2 \):

\[ a^2 = 9801 - 400 \] \[ a^2 = 9401 \]

Taking the square root to find \( a \):

\[ a = \sqrt{9401} \approx 97 \]

So, the missing value that forms a Pythagorean triple with 20 and 99 is approximately 97.

2. For the right triangle with lengths 180 and 181:

Assuming that 181 is the hypotenuse and we want to find the missing side \( a \):

\[ 180^2 + a^2 = 181^2 \]

Calculating the squares:

\[ 32400 + a^2 = 32761 \]

Now, solve for \( a^2 \):

\[ a^2 = 32761 - 32400 \] \[ a^2 = 361 \]

Taking the square root to find \( a \):

\[ a = \sqrt{361} = 19 \]

So, the missing value that forms a Pythagorean triple with 180 and 181 is 19.

Summary of Results

• For the triangle with sides 20 and 99, the missing side is approximately 97.
• For the triangle with sides 180 and 181, the missing side is 19.

Answer 2

first one is wrong

Answer 3

Let’s re-evaluate the first problem step by step carefully.

1. For the right triangle with lengths 20 and 99:

Assuming 99 is the hypotenuse (as it is the larger number) and we are looking for the missing side \( a \):

Using the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

where:

• \( b = 20 \)
• \( c = 99 \)

So the equation becomes:

\[ a^2 + 20^2 = 99^2 \]

Calculating the squares:

\[ a^2 + 400 = 9801 \]

Now, isolate \( a^2 \):

\[ a^2 = 9801 - 400 \] \[ a^2 = 9401 \]

Now taking the square root:

\[ a = \sqrt{9401} \approx 97 \]

Verification

To verify that the calculation is correct, we can ensure:

1. Calculate \( 20^2 \) and \( 99^2 \):

   • \( 20^2 = 400 \)
   • \( 99^2 = 9801 \)

2. Check if \( 400 + 9401 \) equals \( 9801 \):

   • \( 400 + 9401 = 9801 \)

These confirm the calculations were correct.

Therefore, the missing side that works with 20 and 99 is indeed approximately 97.

If you have specific constraints I may have missed or if there was a different expectation for the arrangement of the sides, please clarify, and I would be happy to assist further!

Answer 4

still wrong
A Pythagorean triple consists of three positive integers that work with the Pythagorean Theorem.

Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 20 and 99.

Answer 5

Thank you for your patience. Let’s correctly approach the problem of finding the missing value in a Pythagorean triple formed by two integers 20 and 99.

Given two sides of a right triangle, we can assume that one of them is a leg (let's take 20), and the other can either be a leg or the hypotenuse. The goal is to find a proper integer that makes them a Pythagorean triple.

Case 1: 20 is one leg, 99 is the hypotenuse

Using the Pythagorean theorem, we have:

\[ a^2 + 20^2 = 99^2 \]

Calculating the squares:

\[ a^2 + 400 = 9801 \]

Now, isolate \( a^2 \):

\[ a^2 = 9801 - 400 \] \[ a^2 = 9401 \]

Taking the square root:

\[ a = \sqrt{9401} \approx 96.99 \]

Case 2: 20 is the hypotenuse, and we need to find the other leg

This case is not valid because 20 cannot be the hypotenuse; it's the smaller side.

Conclusion

The first case shows that if 20 is one of the legs and 99 is the hypotenuse, the other leg is not an integer.

To find a suitable integer Pythagorean triple where one side is 20, we can generate or list known triples that include 20. A well-known Pythagorean triple that includes 20 is \( (20, 21, 29) \) (where 29 is the hypotenuse).

Summary

• If considering 99 as the hypotenuse: No integer leg can be found.
• However, if we want a classic Pythagorean triple with 20, it can be verified:

1. For \( (20, 21, 29) \): \[ 20^2 + 21^2 = 29^2 \] \[ 400 + 441 = 841 \quad \text{(True)} \]

So it depends on which configuration is allowable.

If there is a specific triangle configuration or context for 20 and 99 that I am missing, please provide that detail so I can assist you better!