### Video Transcript

Lines ๐ด๐ถ and ๐ต๐ท are the diagonals of a square. The equation of line ๐ต๐ท is ๐ฆ minus three ๐ฅ equals four. Find the equation of line ๐ด๐ถ.

So firstly, what weโre gonna do is actually find the centre of our square. And Iโve labelled the centre of our square ๐ธ in our diagram. We also know that thereโs actually a right angle at ๐ธ. And we have a right angle itโs because we know that ๐ด๐ถ and ๐ต๐ท are the diagonals
of a square. And so, therefore, ๐ด๐ถ and ๐ต๐ท are perpendicular. And this is a relationship thatโs gonna be very useful later on.

So first of all, letโs find the coordinates of point ๐ธ because this is the point
thatโs gonna be on both of our lines because itโs the centre of our square. So straight away, we know that the ๐ฅ-coordinate of point ๐ธ is gonna be zero. So ๐ฅ is equal to zero. And thatโs because itโs on the ๐ฆ-axis.

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

Okay, great! So what do we do now? So now we actually move on to line ๐ด๐ถ because what the question wants us to do is
find the equation of line ๐ด๐ถ. So first of all, to actually find the equation of line ๐ด๐ถ, weโre gonna have a look
at line ๐ต๐ท. So weโve got line ๐ต๐ท is ๐ฆ minus three ๐ฅ is equal to four. Well by actually adding three ๐ฅ to each side, what we can actually do is rearrange
it into ๐ฆ equals three ๐ฅ plus four.

But why would we want to do that? Well, we do that because, actually, it gives it in the form ๐ฆ equals ๐๐ฅ plus
๐. And this is the general form for the equation of a straight line, where ๐ is our
gradient and ๐ is our ๐ฆ intercept. Okay, great! So this is gonna be be really useful.

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

So now itโs actually the gradient thatโs gonna help us find the equation of line
๐ด๐ถ. And thatโs because of the relationship we actually highlighted earlier, because we
said that ๐ด๐ถ and ๐ต๐ท are perpendicular to one another. So theyโre right angles. And we actually know that if two lines are perpendicular to each other, their
gradients, which Iโve called here ๐ one and ๐ two, when theyโre 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 letโs use this to find the equation of line ๐ด๐ถ.

So we can now say that line ๐ด๐ถ, if weโre gonna have it in the form ๐ฆ equals ๐๐ฅ
plus ๐, our ๐, our gradient, is going to be negative a third. And the reason we know itโs negative a third is cause if we look back at the line
๐ต๐ท, we can see that the gradient of the line ๐ต๐ท 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 weโve now found the gradient. What about the ๐ฆ-intercept? Well, from earlier, weโve already found out the ๐ฆ-intercept is equal to four. And thatโs because four is the value of ๐ฆ where the centre of the square is, cause
thatโs where the diagonals actually meet and thatโs on both lines.

So then we can actually prove that by actually substituting our point ๐ธ values into
our equation because weโve got ๐ฆ is equal to negative a third ๐ฅ plus ๐. And we get negative a third ๐ฅ because ๐ is equal to negative a third. So then if we substitute ๐ฆ is equal to four and ๐ฅ 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 weโve said, ๐ is equal to four cause thatโs our ๐ฆ-intercept. So therefore, we can say that the equation of line ๐ด๐ถ is ๐ฆ equals negative a third
๐ฅ plus four.