# Video: AQA GCSE Mathematics Foundation Tier Pack 4 β’ Paper 2 β’ Question 17

Here is a triangle πππ. How long is side ππ? Tick a box. [A] Between 2.5 cm and 7 cm [B] 7 cm [C] Between 7 cm and 9.5 cm [D] More than 9.5 cm

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### Video Transcript

Here is a triangle πππ. How long is side ππ? 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 ππ must be greater than seven centimeters. This is immediately rules out our first two options. The side ππ 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 ππ must also be less than seven centimeters plus 2.5 centimeters. Seven plus 2.5 is equal to 9.5. Therefore, ππ is less than 9.5 centimeters. We can put these two inequalities together such that ππ 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 ππ is between seven centimeters and 9.5 centimeters.

An alternative way to prove this would be using Pythagorasβ theorem. This states that π squared plus π squared is equal to π squared, where π is the length of the longest side or hypotenuse of a right-angle triangle. Substituting in the values from our triangle gives us that ππ 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, ππ squared is equal to 55.25.

In order to calculate the length of ππ, we now need to square root both sides of the equation, as square rooting is the opposite or inverse of squaring. The length ππ 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. ππ is equal to 7.4 centimeters to one decimal place. This confirms that the length of side ππ is between seven centimeters and 9.5 centimeters.

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