How do you know if you can make a right triangle with the square roots line on top of the number like this:
A.2,2, v``4``
B.9,40,41
C.v``5`,10, v``125
What is the methods to find if the numbers can get to a right triangle.
Test for the Pythagorean relationship.
in a right-angled triangle sides a, b, and c, where c is the hypotenuse and obviously the longest side
a^2 + b^2 = c^2
so let's take 2, 2, √4
without even doing anything I can see that it can't be, since √4 = 2 and all sides would be 2
This makes it equilateral, not right-angle
let's look at 9,40,41
clearly 41 is the longest side, so is
9^2 + 40^2 = 41^2
Left side = 81 + 400 = 481
Right sie = 41^2 = 481 , so yes it is right-angled
last one: √5, 10,√125
of those √125 is the longest, so is (√5)^2 + 10^2 = (√125)^2 ?
LS = (√5)^2 + 10^2
= 5 + 100 = 125
RS = (√125)^2 = 125
So yes, it is right-angled
To determine if a set of numbers can form a right triangle, you can use the Pythagorean theorem. The Pythagorean theorem states that 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.
Let's go through each option to see if they form right triangles:
A. (2, 2, √4)
To check if these numbers form a right triangle, we can square each side and see if the sum of the squares of the two shorter sides is equal to the square of the longest side:
2^2 + 2^2 = 4 + 4 = 8
√4^2 = 2^2 = 4
In this case, the sum of the squares of the two shorter sides (8) is not equal to the square of the longest side (4). Therefore, these numbers do not form a right triangle.
B. (9, 40, 41)
Again, we can apply the Pythagorean theorem:
9^2 + 40^2 = 81 + 1600 = 1681
√41^2 = 41^2 = 1681
Here, the sum of the squares of the two shorter sides (1681) is equal to the square of the longest side (1681). Therefore, these numbers do form a right triangle.
C. (√5, 10, √125)
One more time, let's use the Pythagorean theorem to check:
√5^2 + 10^2 = 5 + 100 = 105
√125^2 = 125^2 = 15625
In this case, the sum of the squares of the two shorter sides (105) is not equal to the square of the longest side (15625). Hence, these numbers do not form a right triangle.
So, out of the given options, only option B (9, 40, 41) forms a right triangle.