Video: Pack 4 โ€ข Paper 1 โ€ข Question 23

Pack 4 โ€ข Paper 1 โ€ข Question 23

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Video Transcript

Lines ๐ด๐ถ and ๐ต๐ท are the diagonals of a square. The equation of line ๐ต๐ท is ๐‘ฆ minus three ๐‘ฅ equals four. Find the equation of line ๐ด๐ถ.

So firstly, what weโ€™re gonna do is actually find the centre of our square. And Iโ€™ve labelled the centre of our square ๐ธ in our diagram. We also know that thereโ€™s actually a right angle at ๐ธ. And we have a right angle itโ€™s because we know that ๐ด๐ถ and ๐ต๐ท are the diagonals of a square. And so, therefore, ๐ด๐ถ and ๐ต๐ท are perpendicular. And this is a relationship thatโ€™s gonna be very useful later on.

So first of all, letโ€™s find the coordinates of point ๐ธ because this is the point thatโ€™s gonna be on both of our lines because itโ€™s the centre of our square. So straight away, we know that the ๐‘ฅ-coordinate of point ๐ธ is gonna be zero. So ๐‘ฅ is equal to zero. And thatโ€™s because itโ€™s on the ๐‘ฆ-axis.

So then what we can do is actually substitute this into ๐‘ฆ minus three ๐‘ฅ equals four to actually give us the value of ๐‘ฆ at this point. So therefore, if we do, we get ๐‘ฆ minus three multiplied by zero equals four. So therefore, weโ€™re gonna get ๐‘ฆ is equal to four because ๐‘ฆ minus three multiplied by zero, well three multiplied by zero is just zero. So ๐‘ฆ minus zero equals four. So therefore, ๐‘ฆ must be equal to four. So therefore, we know that the coordinates of the centre of our square, so point ๐ธ, are zero, four.

Okay, great! So what do we do now? So now we actually move on to line ๐ด๐ถ because what the question wants us to do is find the equation of line ๐ด๐ถ. So first of all, to actually find the equation of line ๐ด๐ถ, weโ€™re gonna have a look at line ๐ต๐ท. So weโ€™ve got line ๐ต๐ท is ๐‘ฆ minus three ๐‘ฅ is equal to four. Well by actually adding three ๐‘ฅ to each side, what we can actually do is rearrange it into ๐‘ฆ equals three ๐‘ฅ plus four.

But why would we want to do that? Well, we do that because, actually, it gives it in the form ๐‘ฆ equals ๐‘š๐‘ฅ plus ๐‘. And this is the general form for the equation of a straight line, where ๐‘š is our gradient and ๐‘ is our ๐‘ฆ intercept. Okay, great! So this is gonna be be really useful.

First of all actually, weโ€™ll just go back to something weโ€™ve already done, because if we know that ๐‘ is our ๐‘ฆ-intercept, then therefore it tells us that positive four is gonna be our ๐‘ฆ-intercept. And actually yet if we look back at our point ๐ธ, we can actually see that the ๐‘ฆ-intercept, so the ๐‘ฆ-value where it crosses the ๐‘ฆ-axis, is actually four. So yes that was correct.

So now itโ€™s actually the gradient thatโ€™s gonna help us find the equation of line ๐ด๐ถ. And thatโ€™s because of the relationship we actually highlighted earlier, because we said that ๐ด๐ถ and ๐ต๐ท are perpendicular to one another. So theyโ€™re right angles. And we actually know that if two lines are perpendicular to each other, their gradients, which Iโ€™ve called here ๐‘š one and ๐‘š two, when theyโ€™re multiplied together are equal to negative one.

And another way we can actually say that is that actually the gradient of a line is the negative reciprocal of the gradient of the line that is perpendicular to it. Okay, great! So letโ€™s use this to find the equation of line ๐ด๐ถ.

So we can now say that line ๐ด๐ถ, if weโ€™re gonna have it in the form ๐‘ฆ equals ๐‘š๐‘ฅ plus ๐‘, our ๐‘š, our gradient, is going to be negative a third. And the reason we know itโ€™s negative a third is cause if we look back at the line ๐ต๐ท, we can see that the gradient of the line ๐ต๐ท was three. And therefore, negative a third multiplied by three is gonna give us negative three over three, which gives us negative one. And again, using the other definition, we can see that, well, negative one over three or negative a third is actually that negative reciprocal of three.

Okay, great! So weโ€™ve now found the gradient. What about the ๐‘ฆ-intercept? Well, from earlier, weโ€™ve already found out the ๐‘ฆ-intercept is equal to four. And thatโ€™s because four is the value of ๐‘ฆ where the centre of the square is, cause thatโ€™s where the diagonals actually meet and thatโ€™s on both lines.

So then we can actually prove that by actually substituting our point ๐ธ values into our equation because weโ€™ve got ๐‘ฆ is equal to negative a third ๐‘ฅ plus ๐‘. And we get negative a third ๐‘ฅ because ๐‘š is equal to negative a third. So then if we substitute ๐‘ฆ is equal to four and ๐‘ฅ is equal to zero into that, we get four is equal to negative third multiplied by zero plus ๐‘. So therefore, we get four is equal to ๐‘.

So as weโ€™ve said, ๐‘ is equal to four cause thatโ€™s our ๐‘ฆ-intercept. So therefore, we can say that the equation of line ๐ด๐ถ is ๐‘ฆ equals negative a third ๐‘ฅ plus four.

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