We are given four sets of numbers that represent the side measures of a triangle. We want to identify the set that does not belong with the other three. If we know the side measures of a triangle, we can check if it is a right triangle. Let's check which of the given measures represent a right triangle. We can use the Converse of the Pythagorean Theorem to help us.
Converse of the Pythagorean Theorem
If the sides of a triangle have lengths a, b, and c and c^2=a^2+b^2, then the triangle is a right triangle.
This tells us that we can use the Pythagorean Theorem in reverse to test if a triangle with the given side lengths is right. The hypotenuse c has the greatest value of the lengths of a right triangle. Let's substitute the side lengths from each set into a^2+b^2=c^2 and see if they produce a true statement.
Set
a
b
c
a^2+b^2=c^2
Simplify
True or False?
3 - 4 - 5
3
4
5
3^2+ 4^2? = 5^2
25 = 25
True
12 - 35 - 37
12
35
37
12^2+ 35^2? = 37^2
1369 = 1369
True
3 - 5 - 7
3
5
7
3^2+ 5^2? = 7^2
34 ≠49
False
6 - 8 - 10
6
8
10
6^2+ 8^2? = 10^2
100 ≠100
True
Notice that the measures from the third set produced a false statement, while the first, second, and fourth sets produced a true statement. This means that the third set of numbers are not measures of the sides of a right triangle, while the other sets are. Now we can see that the third set does not belong with the other three sets.
