How do you use the Pythagorean Theorem to determine if the three numbers could be the measures of the sides of a right triangle: 6, 12, 18?

Question

Julia Adams · Accepted Answer

Check the similar triangle with sides #1#, #2#, #3# against Pythagoras formula to find that this is not a right angled triangle.

Three positive numbers can be the measures of the sides of a right triangle if and only if taken in ascending order they satisfy: #a^2+b^2=c^2# Also a triangle is a right triangle if and only if any similar triangle is a right triangle. So you can multiply or divide #a#, #b# and #c# by any non-zero number before applying the test. In our example, all of the sides are divisible by #6# so let us assign: #a = 6/6 = 1# #b = 12/6 = 2# #c = 18/6 = 3# We find #a^2+b^2 = 1^2+2^2 = 1+4 = 5 != 9 = 3^2 = c^2# In fact these side lengths only form a degenerate 'triangle' of zero area, with interior angles #0#, #0# and #pi#

Eva Abramson · Answer

undefined To determine if the three numbers 6, 12, and 18 could be the measures of the sides of a right triangle, we can use the Pythagorean Theorem. According to the theorem, 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.

Using the Pythagorean Theorem, we can calculate the squares of the lengths of the sides: 
6^2 = 36
12^2 = 144
18^2 = 324

Next, we check if the sum of the squares of the two shorter sides is equal to the square of the longest side (hypotenuse). In this case, 36 + 144 = 180, which is not equal to 324.

Therefore, the three numbers 6, 12, and 18 cannot be the measures of the sides of a right triangle, as they do not satisfy the Pythagorean Theorem.