Can you solve this clever problem ?
The problem we are going to see today is a great example of Looks can be deceptive. You will understand what I mean by this as we go along.
References:
• There is a ridiculously easy way to solve ...
Script:
Hello world and welcome to another funza academy math video.
Now before looking at the problem, I ask you to please subscribe to this channel.
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Ok, now the problem.
So we have a triangle.
Now at a distance of 5 cm from each vertex we draw a straight line, like so.
After doing so, this is what we get.
Now, imagine, we chop these 3 triangles. The resulting figure we get is this .
This figure has 6 sides. In geometry we call such a figure - a hexagon.
So here is our question - What is the area of this hexagon?
And here are the lengths of the sides of the hexagon
And here are all three figures, side by side, with all the lengths.
So please pause the video and give the problem a shot.
And when you are done, please return and continue watching the solution.
Good luck.
Welcome back.
Now here are some thoughts that can come to mind:
Can I directly calculate the are of the hexagon using some formula?
Can I indirectly calculate the area of the hexagon?
Let us focus on the second option.
The area of the hexagon is the area of the overall triangle minus the area of the 3 smaller triangles.
But what is the area of the overall triangle?
Now as you know triangles are of different types - right angle triangles, equilateral triangles, isosceles triangles and so on.
By just looking at the figure, the triangle does not look like either a right angle or an equilateral triangle.
But let us be sure since knowing that may make our life easy.
So the larges side is 25 and the other 2 sides are 15 and 20.
So lets square the numbers
So we have 625, 225 and 400
and using Pythagoras theorem we see that indeed 625 = 225 + 400.
How interesting - Our figure was not drawn to scale so was misleading. But we were smart to not fall into that trap.
Let me also re-draw the over all figure to represent what it is , a right angled triangle.
So now that our overall triangle is a right angle triangle, we can easily find 2 areas, yes 2 - this overall triangle and this smaller one.
Now that leaves us with these 2 other triangles.
Now , please pay attention to these points.
Our overall triangle is a right angled triangle of sides 25, 20 and 15.
It is a scaled up version of the 3, 4, 5 right angle triangle.
So what does this mean?
This means that if we drop a perpendicular line from this point to this base, the length of the perpendicular will be 3 and this length 4.
As you can see, another 3, 4,5 right angle triangle.
And if we perform the same operation here, dropping a perpendicular to this base, the length the perpendicular is 4 and this length, 3.
So another 3,4,5 right angle triangle.
And this means that we can determine the areas of the 2 triangles here.
Now guys, at this point, if things are not clear, I request you to bring out your ruler and draw a right angle triangle with sides 25, 20 and 15 and try this out. That will surely help
Ok , now lets hit the home stretch.
The area of our overall triangle is -
The are of this triangle is
This triangle is
This triangle is
and finally this triangle is
So this means that the area of our hexagon is the area of the larger triangle minus the areas of each smaller triangle.
So the area of our hexagon is
So then, this brings us to the end of another video.
I hope you enjoyed it as much as i enjoyed making it.
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Take care and see you next week