Triangle 𝑂𝑀𝑁 has a perimeter of
30 centimeters. 𝑂𝑀 equals 12 centimeters. 𝑂𝑁 equals 13 centimeters. Determine by calculation where
triangle 𝑂𝑀𝑁 is a right-angled triangle.
The perimeter of any triangle is
the sum of its three lengths. In this case, 𝑂𝑀 plus 𝑂𝑁 plus
𝑀𝑁 is equal to 30 as the perimeter is 30 centimeters. Substituting in the length of 𝑂𝑀
and 𝑂𝑁 gives us 12 plus 13 plus 𝑀𝑁 is equal to 30. 12 plus 13 is equal to 25. Finally, subtracting 25 from both
sides of this equation gives us a length of 𝑀𝑁 of five centimeters.
As we now know all three lengths of
the triangle, we can check whether it is right angled by using Pythagoras’s
theorem. This states that in right-angled
triangles 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the
hypotenuse or longest side of the right-angled triangle.
Squaring the two shorter sides 𝑂𝑀
and 𝑀𝑁 gives us 12 squared plus five squared. 12 squared is equal to 144 and five
squared is equal to 25. Adding these two values give us
169. The hypotenuse is 𝑂𝑁. Therefore, we need to square
13. 13 squared is also equal to
We can, therefore, say yes, 𝑂𝑀𝑁
is a right-angled triangle as it satisfies Pythagoras’s theorem. Any triangle that satisfies
Pythagoras’s theorem will be right angled.