Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 2 • Question 12

Which of the following points is exactly 13 units away from the point (0, 0)? Circle your answer. [A] (1, 12) [B] (5, 12) [C] (5, 8) [D] (−1, 14)

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Video Transcript

Which of the following points is exactly 13 units away from the point zero, zero? Circle your answer. The possibilities are one, 12; five, 12; five, eight; or negative one, 14.

When we say that a point is exactly 13 units away from the point zero, zero, the origin, we mean that the straight line distance between these two points is 13 units. Suppose this point has coordinates 𝑥, 𝑦. If we draw in a right-angled triangle below this line, then we can see the relationship that exists between 𝑥, 𝑦 and this length of 13.

The horizontal side of the triangle, the base, will be 𝑥 units because this is the difference between the 𝑥-coordinates of the two points: 𝑥 minus zero which is just 𝑥. The vertical side of the triangle, its height, will be 𝑦 units because this is the difference between the 𝑦-coordinates: 𝑦 minus zero which is just 𝑦.

As we have a right-angled triangle, we can apply Pythagoras’s theorem, which tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse — the longest side. So we have that 𝑥 squared plus 𝑦 squared equals 13 squared. 13 squared is equal to 169. So we can rewrite this as 𝑥 squared plus 𝑦 squared equals 169.

The question now is for which of these coordinates is 𝑥 squared plus 𝑦 squared equal to 169. So we can take each of them in turn. For the first bit, 𝑥 squared plus 𝑦 squared is one squared plus 12 squared which is one plus 144. This is equal to 145 not 169. So the point one, 12 is not exactly 13 units away from the point zero, zero.

Next, we consider the point five, 12. And this time, we have five squared plus 12 squared which is equal to 25 plus 144. 25 plus 144 is equal to 169 which means that point five, 12 is exactly 13 units away from the point zero, zero. We found then one point that does work. But the question doesn’t explicitly say that there’s only one answer. So we need to check the two remaining points.

For the point five, eight, we get five squared plus eight squared which is equal to 25 plus 64 which is 89. So this point is not exactly 13 units away from the point zero, zero.

For the final point, negative one, 14, we have to be a little bit careful because one of the values is negative. When we square negative one, this means negative one multiplied by negative one which is positive one because a negative multiplied by a negative gives a positive. So we have one plus 196 which is equal to 197 not 169, which tells us that this point is not exactly 13 units away from the point zero, zero.

So the only one of the points which is exactly 13 units away from the origin is the point five, 12. You may actually have recognized this straightaway because five, 12, 13 is an example of a Pythagorean triple. That’s a right-angled triangle in which all three sides are integers.

If you’re familiar with Pythagorean triples, then you may have been able to spot straightaway that the point five, 12 is exactly 13 units away from the point zero, zero.

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