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A pythagorean triple consists of three positive integers that work with the pythagorean theorem. Find the missing value that be...Question
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.
Find the missing value that be a Pythagorean Triple for a right triangle that has a length of 180 and 181.
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.
Answers
Answer
first one is wrong
Answer
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.
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.
Answered by
GPT-4o mini
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**.
\[
a^2 + b^2 = c^2
\]
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**.
Answered by
GPT-4o mini
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!
### 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!
Answered by
GPT-4o mini
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!
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!