Here is a triangle 𝑋𝑌𝑍. How long is side 𝑋𝑍? Tick a box. Is it between 2.5 centimeters and seven centimeters? Is it seven centimeters? Is it between seven centimeters and 9.5 centimeters? Or is it more than 9.5 centimeters?
The first thing to note about our triangle is that it is right-angled. The side opposite the right angle is called the hypotenuse. This is the longest side in any right-angle triangle. This means that, in our question, it must be greater than seven centimeters and 2.5 centimeters. As seven is greater than 2.5, we can say that 𝑋𝑍 must be greater than seven centimeters. This is immediately rules out our first two options. The side 𝑋𝑍 cannot be between 2.5 and seven centimeters and cannot be equal to seven centimeters.
In any triangle, no side can be equal to or greater than the sum of the other two sides. This means that 𝑋𝑍 must also be less than seven centimeters plus 2.5 centimeters. Seven plus 2.5 is equal to 9.5. Therefore, 𝑋𝑍 is less than 9.5 centimeters. We can put these two inequalities together such that 𝑋𝑍 is greater than seven centimeters and less than 9.5 centimeters. It cannot be more than 9.5 centimeters. Therefore, the correct answer is that 𝑋𝑍 is between seven centimeters and 9.5 centimeters.
An alternative way to prove this would be using Pythagoras’ theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse of a right-angle triangle. Substituting in the values from our triangle gives us that 𝑋𝑍 squared is equal to seven squared plus 2.5 squared. Seven squared is equal to 49 as seven multiplied by seven is 49. And 2.5 squared is equal to 6.25. Adding these two numbers gives us 55.25. Therefore, 𝑋𝑍 squared is equal to 55.25.
In order to calculate the length of 𝑋𝑍, we now need to square root both sides of the equation, as square rooting is the opposite or inverse of squaring. The length 𝑋𝑍 is equal to the square root of 55.25. Typing this into the calculator gives us 7.43 and so on. As three is less than five, we can round down if rounding to one decimal place. 𝑋𝑍 is equal to 7.4 centimeters to one decimal place. This confirms that the length of side 𝑋𝑍 is between seven centimeters and 9.5 centimeters.