Determine whether the triple of numbers can be the sides of a right triangle. Please explain.
6, 8, 10
Squareroot of 2, Squareroot of 3, Squareroot of 10
If it is a right-angled triangle, then clearly the largest number would have to be the hypotenuse.
so test if 6^2 + 8^2 = 10^2 ?
Yes it is, so the 3 sides form a right-angled triangle.
is (√2)^2 + (√3)^2 = (√10)^2 ?
What do you think?
What is your conclusion?
To determine whether a triple of numbers can be the sides of a right triangle, we need to check if the Pythagorean theorem holds true for the given numbers. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's evaluate the first triple of numbers:
6, 8, and 10.
According to the Pythagorean theorem:
Hypotenuse squared = a^2 + b^2
If we assume 6 and 8 to be the two sides of the right triangle, we can calculate the length of the hypotenuse:
Hypotenuse squared = 6^2 + 8^2
Hypotenuse squared = 36 + 64
Hypotenuse squared = 100
Now, if we take the square root of 100, we get the length of the hypotenuse:
Hypotenuse = √100 = 10
Since the calculated length of the hypotenuse matches the given length of the third side (10), the triple of numbers (6, 8, 10) can be the sides of a right triangle.
Now let's evaluate the second triple of numbers:
Square root of 2, Square root of 3, Square root of 10.
To check whether these numbers can form a right triangle, we need to square each of them:
(√2)^2 = 2
(√3)^2 = 3
(√10)^2 = 10
According to the Pythagorean theorem:
Hypotenuse squared = a^2 + b^2
If we assume √2 and √3 to be the two sides of the right triangle, we can calculate the length of the hypotenuse:
Hypotenuse squared = (√2)^2 + (√3)^2
Hypotenuse squared = 2 + 3
Hypotenuse squared = 5
However, the third side (√10) has a squared length of 10, which does not match the calculated length of the hypotenuse (5). Therefore, this triple of numbers (Square root of 2, Square root of 3, Square root of 10) cannot be the sides of a right triangle.
By applying the Pythagorean theorem and comparing the calculated lengths, we can determine whether a triple of numbers can form the sides of a right triangle.