Square Roots And Pythagorean Theorem

Explanation
The two square roots of 64 are 8 and -8 because when a number is squared, it gives the original number. In this case, 8 squared is 64, and -8 squared is also 64. Therefore, both 8 and -8 are valid square roots of 64.

Explanation
The square root of 81 is both 9 and -9 because when a number is squared, the result is always positive. Therefore, the square root of a positive number has two solutions, one positive and one negative. In this case, both 9 and -9, when squared, equal 81.

Explanation
The number 1 is a perfect square because it can be expressed as the square of 1, which is 1.

Explanation
The square root of 10 is approximately between 3 and 4 because when we square 3, we get 9 which is less than 10, and when we square 4, we get 16 which is greater than 10. Therefore, the square root of 10 lies between these two numbers.

Explanation
The square root of 256 is -16. This is because the square root of a number is a value that, when multiplied by itself, gives the original number. In this case, -16 multiplied by -16 equals 256.

Explanation
To approximate the square root of 161 between 12 and 13, we can start by finding the square root of the nearest perfect square to 161, which is 144 (12^2). The difference between 161 and 144 is 17. Since 17 is closer to the lower bound (12) than the upper bound (13), we can estimate that the square root of 161 is slightly greater than 12. Therefore, the answer is between 12 and 13.

Explanation
The hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the squares of the legs are 5^2 = 25 and 6^2 = 36. Adding these together gives 61. Taking the square root of 61 gives approximately 7.8, which is the length of the hypotenuse.

Explanation
You can use the Pythagorean Theorem to find the length of the hypotenuse (the side opposite the right angle) of a right triangle when you know the lengths of the two legs. The Pythagorean Theorem states that in a right triangle:
a² + b² = c²
Where:
"a" and "b" are the lengths of the legs.
"c" is the length of the hypotenuse.
In your case, the legs are 30 and 40. Plug these values into the formula:
30² + 40² = c²
900 + 1600 = c²
2500 = c²
Now, take the square root of both sides to find "c," the length of the hypotenuse:
c = √2500
c = 50
So, the length of the hypotenuse of the right triangle is 50 units.

Explanation
The given answer is correct because all the listed sets of numbers have the potential to be side lengths of right triangles. In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean theorem. By applying this theorem to each set of numbers, it can be determined that all of them satisfy the conditions and can form right triangles.