Draw a triangle with 𝐵𝐶 equals
seven centimetres, angle 𝐵 equals 45 degrees, and angle 𝐴 equals 105 degrees. Then construct another triangle
whose sides are three-quarters times the corresponding side of triangle 𝐴𝐵𝐶.
Now, let’s plan our attack before
we do any constructions. We’re gonna have a triangle
𝐴𝐵𝐶. So this doesn’t have to be
accurate. And side 𝐵𝐶 is gonna be seven
centimetres. Angle 𝐵 equals 45 degrees and
angle 𝐴 is 105 degrees.
Now, angles in a triangle sum to
180 degrees. This means that angle 𝐶 is 180
minus 45 minus 105. That’s 30 degrees. So, it’s gonna be easier to do our
drawing if we draw the baseline of seven centimetres first, measure a line up at 45
degrees, measure another line up at 30 degrees, and then they will intersect at
Now, with all constructions, I
recommend that you get yourself a nice sharp pencil, a nice clear plastic ruler with
millimetre markings, a pair of compasses. Now, this is an accurate pair of
bought compasses. You could just as well use a very
cheap pair of compasses. But do remember to tighten that
screw so that the radius stays where you put it.
Now, when you’re doing drawings,
you’re also allowed to use a protractor to measure your angles. Okay, let’s start by drawing 𝐵𝐶
with length seven centimetres. Let’s put down our ruler, mark off
the zero line, mark off the seven line — so we know where the end of the lines are —
and then join them up. This is point 𝐵 and this is point
Now, we need angle 𝐶𝐵𝐴 to be 45
degrees. Let’s bring in the protractor, line
up the baseline of the protractor with the line 𝐵𝐶, and line up the cross here
with point 𝐵. Do this nice and accurately. Then, start counting from zero up
to 45 degrees. Now, put your pencil at point 𝐵,
bring in the ruler, swivel it round, and draw in that line.
Now, we need to make angle 𝐵𝐶𝐴
30 degrees. Again, bring the protractor. Make sure the baseline is on line
𝐵𝐶. Make sure the cross here is lined
up exactly with point 𝐶. Start counting from zero the
outside numbers and go to 30 degrees. Now pencil on point 𝐶, bring in
the ruler, swivel it up to that mark you’ve just made, and draw in the line. Where they intersect is point
𝐴. Now, let’s just check that. It should be 105 degrees. Starting at zero, going all the way
up, 105 degrees, looks like we’re right.
Now, we’re gonna construct another
triangle whose sides are only three-quarters times the corresponding sides of
triangle 𝐴𝐵𝐶. And we got to construct this
triangle. We can’t just measure the sides and
use our ruler to measure the new sides. The way to do this is to put your
pencil at point 𝐵 and draw a line going down at an acute angle. It doesn’t exactly matter what that
angle is, but something like that.
Now, we want our sides to be
three-quarters the times of the corresponding side. So if I draw four marks down here
which are equidistant, we can use that to do a construction. So bring any compasses, make sure
that they’re open less than a quarter of the distance of this, so very slightly
narrower, and just mark off those arcs.
So put your compass point at 𝐵,
mark off an arc. Move the compass point to where the
arc was, mark off another one, and again, and again. Now, it’s important that you kept
your radius exactly the same for each of those. These must be equidistant.
Let’s label those points: 𝑥 one,
𝑥 two, 𝑥 three, and 𝑥 four. Now, we can draw our line from 𝑥
four up to point 𝐶. Now, these points chop this line
into four equal parts. If I draw a line which is parallel
to 𝑥 four 𝐶, running from point 𝑥 three up to 𝐵𝐶 where it intersects 𝐵𝐶, will
be three-quarters of the way along 𝐵𝐶.
So I need to replicate this angle
here, over here. Now, the way to do this is to get a
compasses, put the point at 𝑥 four, and draw an arc between 𝐵 𝑥 four and 𝐶 𝑥
four. Keeping the radius exactly the
same, put the compass point at 𝑥 three and replicate that arc.
Now, we need to measure the
distance from the point of intersection on this line to the point of intersection on
this line. So we can do this with the
compasses. Just open them up nice and
accurate. Now, we can replicate that distance
from this arc. So that distance is the same as
Now, this line goes 𝑥 four through
this point of intersection up to 𝐶. So if we draw the line from 𝑥
three through that point of intersection up to 𝐵𝐶, it will be parallel to the
original line. So that line is parallel with that
line because 𝑥 three is three-quarters of the way along 𝐵 𝑥 four. Then, this point here — let’s call
it 𝐶 dash — will be three-quarters of the way along 𝐵𝐶.
Now, we need to do the same from
this point up to here doing a line that’s parallel with 𝐴𝐶. So we want to replicate this angle
over here. Let’s bring our compass point in to
𝐶 and draw an arc that intersects those two lines. Move the compass point to point 𝐶
dash and replicate that arc.
Now, measure the distance between
where the arc intersects 𝐵𝐶 and where it intersects 𝐴𝐶. We can use the compasses to do this
by putting the point on here and then closing up that distance until it exactly
matches. That looks like there. Now, we can replicate that distance
by putting the compass point here.
And as before, this line 𝐶 to 𝐴
went up through that point of intersection. If we go from 𝐶 dash up to 𝐴𝐵
through that point of intersection, then it will be parallel with the line above
it. So this point up here is 𝐴
Now, we can just make our triangle
a little bit bolder. And it’s important you don’t rub
out any of your construction lines because this is your working out.