### Video Transcript

A quadrilateral has vertices at the
point π΄, π΅, πΆ, and π·, which have coordinates negative two, two; two, six; three,
three; and one, one, respectively. πΈ is the midpoint of π΄π΅. And πΉ is the midpoint of πΆπ·. Prove that lines π΄π΅ and πΈπΉ are
perpendicular. You must explain every step of your
working out.

Two lines are perpendicular if
their gradients, given here as π one and π two, have a product of negative
one. In this case then, we need to work
out the gradients of the lines π΄π΅ and πΈπΉ. Remember, the formula for gradient
is change in π¦ divided by change in π₯, which is sometimes written as π¦ two minus
π¦ one all over π₯ two minus π₯ one.

To find the gradient of the line
π΄π΅, we therefore need to substitute the coordinates of π΄ and π΅ into this
formula. Change in π¦ is six minus two. And change in π₯ is two minus
negative two.

Remember, itβs absolutely fine to
perform this calculation the other way round as long as youβre consistent. If you do two minus six for change
in π¦, you then have to do negative two minus two for change in π₯. Six minus two is four. And two minus negative two, which
becomes two plus two, is also four. Four divided by four is one. The gradient of the line passing
between π΄ and π΅ is, therefore, one.

Now before we can work out the
gradient of the line πΈπΉ, we first need to work out the coordinates of both πΈ and
πΉ. πΈ is the midpoint of π΄π΅. And πΉ is the midpoint of πΆπ·. To find the midpoints, we add
together the π₯-coordinates and divide by two and we add together the π¦-coordinates
and divide by two.

Remember, πΈ is the midpoint of π΄
and π΅. So its π₯-coordinate is negative
two plus two all over two. And its π¦-coordinate is two plus
six, again all over two. Negative two plus two is zero. So dividing it by two, we still get
zero. And two plus six divided by two is
eight divided by two, which is four.

We can find the coordinates for πΉ
by repeating this process for points πΆ and π·. In this case, three plus one over
two is the π₯-coordinate and three plus one over two is also the π¦-coordinate. Three plus one is four, which
divided by two is two. So the coordinate for πΉ is two,
two.

Once we have these coordinates, we
can then find the gradient of the line passing through πΈπΉ using the formula from
before. Change in π¦ is two minus four. And change in π₯ is two minus
zero. Two minus four is negative two. And negative two divided by two is
negative one. We found the gradient of the line
πΈπΉ to be negative one.

Now remember, we said that two
lines are perpendicular if the product, which means multiply, of their gradients is
negative one. The gradient of π΄π΅ is one and the
gradient of πΈπΉ is negative one. One multiplied by negative one is
negative one. Donβt forget to include a
conclusion. π΄π΅ and πΈπΉ are
perpendicular.