# Video: Solving for One of the Legs of a Right-Angled Triangle with Integer Solutions

Given that 𝐴𝐵𝐶 is a right-angled triangle, where 𝐶𝐵 = 8 cm and 𝐴𝐶 = 10 cm, determine the length of 𝐴𝐵.

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### Video Transcript

Given that 𝐴𝐵𝐶 is a right-angled triangle, where 𝐶𝐵 equals eight centimetres and 𝐴𝐶 equals 10 centimetres, determine the length of 𝐴𝐵.

As our triangle is right-angled, we can use Pythagoras’s theorem. This states that the square of the length of a triangle’s hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, 𝐴𝐵 squared plus 𝐶𝐵 squared equals 𝐴𝐶 squared.

If we let the length 𝐴𝐵 equal 𝑥 and substitute the values into Pythagoras’s theorem. We get 𝑥 squared plus eight squared equals 10 squared. Eight squared is equal to 64 as eight multiplied by eight is 64. 10 squared is equal to 100. Subtracting 64 from both sides of the equation leaves us with 𝑥 squared equals 36. And finally, square rooting both sides of the equation gives us 𝑥 equals six as the square root of 𝑥 squared is 𝑥 and the square root of 36 is equal to six.

Therefore, the length 𝐴𝐵 in the right-angled triangle is equal to six centimetres. Pythagoras’s theorem can be used to find the missing length of any right-angled triangle.