### Video Transcript

Find the length of the
perpendicular drawn from the origin to the straight line negative three π₯ plus four
π¦ minus 21 equals zero rounded to the nearest hundredth.

So within this question, weβve been
given a point, the origin, and weβve been given the equation of a straight line:
negative three π₯ plus four π¦ minus 21 equals zero. Weβre asked to find the length of
the perpendicular between these two. So that is the length of the
perpendicular line that starts at the origin and is drawn to meet the straight
line.

So in order to answer this
question, we need to recall that in fact we have a standard formula for answering
questions like this. If we want to calculate the
perpendicular distance between a point with coordinates π₯ one, π¦ one and a line
with equation ππ₯ plus ππ¦ plus π is equal to zero, then we have the formula π,
which represents the length, is equal to the modulus or absolute value of ππ₯ one
plus ππ¦ one plus π all divided by the square root of π squared plus π
squared.

Those vertical lines in the
numerator of the formula represent the modulus or absolute value, which means the
size of a number ignoring its sign. So essentially, this means its
distance from zero. The modulus of seven is seven, but
the modulus of negative seven is also seven. So what we need to do for this
question is determine the values of π, π, π, π₯ one, and π¦ one and then
substitute them into the standard formula.

π₯ one, π¦ one first of all
represents the coordinates of the point that weβre drawing this perpendicular
from. And in our question, this point is
the origin. The origin has the coordinates
zero, zero. And therefore, the values of π₯ one
and π¦ one are both zero. In order to find the values of π,
π, and π, we need to look at the equation of the line: negative three π₯ plus four
π¦ minus 21 is equal to zero.

So comparing these two forms, we
see that π is equal to negative three, π is equal to four, and π is equal to
negative 21. Now we have all the information we
need in order to be able to use the formula. So we need to substitute each of
the values in the correct places.

So in the numerator first of all,
we have π is equal to the modulus of π multiplied by π₯ one, thatβs negative three
multiplied by zero, plus π multiplied by π¦ one, so thatβs four multiplied by zero,
and then plus π, so thatβs plus negative 21. Then we divide by the square root
of π squared plus π squared, so thatβs the square root of negative three squared
plus four squared. So this simplifies to the modulus
of negative 21 over the square root of nine plus 16.

Now, the modulus of negative 21,
remember thatβs the size of the number ignoring its sign. So the modulus of negative 21 is
just 21. In the denominator, nine plus 16 is
25. So we have 21 over the square root
of 25. 25 is a squared number, and it has
an exact square root, which is five. So our answer simplifies to 21 over
five. As a decimal, 21 over five is equal
to 4.2.

Now the question has actually asked
us to give our answer to the nearest hundredth. And our answer is currently to the
nearest tenth, so we just need to add an extra zero in order to give our answer in
the requested format. So we have the length of the
perpendicular between the origin and the specified line is 4.20 length units. We use length units here because we
donβt know the scale of the coordinate grid, so we canβt say that itβs centimetres
or millimetres or so on. We have to use the general length
units.