Video: CBSE Class X • Pack 2 • 2017 • Question 26

CBSE Class X • Pack 2 • 2017 • Question 26


Video Transcript

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 𝐴.

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 𝐶.

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 that distance.

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 dash.

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.

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