Draw a triangle 𝐴𝐵𝐶 where 𝐴𝐵 equals six centimeters, angle 𝐴 equals 30 degrees,
and angle 𝐵 equals 60 degrees. Construct another triangle 𝐴𝐵 dash 𝐶 dash similar to triangle 𝐴𝐵𝐶, with base
𝐴𝐵 dash equals eight centimeters.
Now, this is worth doing a quick sketch before we start just so we can plan our
approach. So we’ve got triangle 𝐴𝐵𝐶, where 𝐴𝐵 is six centimeters, angle 𝐴 is 30 degrees,
and angle 𝐵 is 60 degrees. Now, we got to construct another triangle with the same point 𝐴, a point 𝐵 dash
which is over here somewhere, and point 𝐶 dash which is over here somewhere with
base 𝐴𝐵 dash equals eight centimeters.
And this needs to be a similar triangle. So, angle 𝐴 still needs to be 30 degrees, angle 𝐵 dash needs to be 60 degrees, and
angle 𝐶 dash will still be 90 degrees. So 𝐶𝐵 and 𝐶 dash 𝐵 dash will be parallel. Now, let’s think about this. 𝐴𝐵 dash over 𝐴𝐵 is eight over six, which is equal to four over three.
So the dimensions in the larger triangle are gonna be four-thirds of the dimensions
in the smaller triangle or the scale factor is gonna be four-thirds. Okay, let’s draw our triangle 𝐴𝐵𝐶. We need a baseline of six centimeters. Angle 𝐴 needs to be 30 degrees. Bring the protractor in, line up the baseline along 𝐴𝐵, make sure the crosshair
lines up at point 𝐴, and then start our counting from zero up to 30 degrees.
Angle 𝐵 needs to be 60 degrees. Okay, bring in the protractor, line up the baseline, line up the crosshair, and then
start counting from zero up to 60 degrees, where the intersect will be point 𝐶. So let’s extend line 𝐴𝐵. And 𝐵 dash is gonna be out here somewhere.
Now because we got to do a construction, we’re not just allowed to measure eight
centimeters on here. We may be able to do that at the end to check our answer. But we are not allowed to do that now. What we’re gonna do is draw a line from 𝐴 at an acute angle to 𝐴𝐵 coming down
here. Now, the exact distance of that line doesn’t really matter.
Now, remember 𝐴𝐵 dash is four-thirds of 𝐴𝐵 or 𝐴𝐵 is three-quarters of 𝐴𝐵
dash. So if I make four equidistant marks down here and then we take the third one and make
a line up to 𝐵 then draw a line parallel with that, this will give us where point
𝐵 dash is.
So let’s bring in our compasses. They don’t need to be too big cause you’re gonna make four equidistant arcs all the
way down here. So let’s do that. One: put point here; two: point here; three; and again four. Let’s label those points 𝑥 one, 𝑥 two, 𝑥 three, and 𝑥 four. And let’s join our point 𝑥 three to point 𝐵.
Now, if we can replicate that angle at 𝑥 four and then have this line going up here,
that will be parallel to 𝑥 three 𝐵. And then, I’ll mark off a point which is four-thirds of the distance of 𝐴𝐵. Look 𝐴 𝑥 four is four-thirds of 𝐴 𝑥 three. Let’s open them up and draw an arc between 𝐴 𝑥 three and 𝐵 𝑥 three.
Now, let’s duplicate that arc from point 𝑥 four. Now, if we can work out the distance between where it insects this line and where it
intersects this line and replicate that on here, we’ll know where to draw that
line. So, put your compass point here, open them up nice and accurately to work out that
distance, and then replicate that distance on here. That’s because the line from 𝑥 three through that intersection up to 𝐵 goes
If we do the same here, 𝑥 four through that intersection up to this point, up to
this line here, we’ll have our point 𝐵 dash. Pencil on 𝑥 four, swivel the ruler around, draw through the intersection. And this is 𝐵 dash. And this line is parallel to this line.
Now, we need to do the same here. We need to replicate this line over here. So we got to get this angle and recreate it over here. Let’s bring in the compasses, put the point at 𝐵, and let’s close that up a little
and create an arc. Then, use the same radius to create that arc at 𝐵 dash.
Now, we need to measure the distance between here and here, which we can do by
placing the compass point here and then opening up the compasses until they
cross. So that’s look the same. Now, we can replicate that distance here. And remember 𝐵 through the intersection up to the line 𝐴𝐶, we’re gonna get 𝐵 dash
through the intersection up to that line past 𝐴𝐶.
Pencil point on 𝐵, bring it up, swivel the ruler through that point there up to
here. And this will be point 𝐶 dash. Now, this line is parallel to this line. And this angle is the same as this angle. Now, let’s just make our triangle 𝐴𝐵 dash 𝐶 dash a bit bolder to make it stand
out. And there, we have it.
Now, we said we do one final check: 𝐴𝐵 dash should be eight centimeters. Let’s measure this. This is the moment of truth on our ruler. Yup! That looks like that’s eight centimeters. It looks like we’ve done nice, accurate constructions.
Lastly, it’s important to say, “Don’t rub out any of these construction lines. This is your working out. It’s very important.”