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This question is really fun to solve. It
asks us to calculate the shaded area
which is this area here. We know that
this region has 30 unit squares of area.
This other one has 34 and this one has
42. We also know that this piece has the
same size as this one which has the same
size as this one and the same size as
this one and this and this and this and
this. That is all pieces are equal. In
other words, they come exactly from the
midpoint of each side of the square. So,
can you solve it? Well, the question
asks for the shaded area. My question is
simple. What figure is this? You might
say, "Hey, brain station, it's a
quadrilateral because it has four
sides." All right, I agree. Now, how do
we calculate the area of that thing?
Like, how do I find the area of this
quadrilateral? That's hard to answer,
right? So, in order to make my life
easier, I'm going to turn this shape
into a figure whose area I do know how
to calculate. And what figure would that
be? A triangle. If you thought of a
triangle, then congrats. You are right.
So, I'm going to divide this figure here
into two triangles like this. And I know
how to calculate the area of a triangle.
Its base time height / 2, right? But I'm
not going to do that with just this one.
I'm going to do the same with all these
quadrilaterals.
So look here. I will divide this into
two triangles. Then I will divide here
too into two triangles. And I will not
spare you as well. Perfect. And now what
do we do after that? What's the next
step? Come on, think. Let's stop looking
at this part here which is what the
question is asking and let's look at the
rest of the figure so we can start
thinking outside the box. Let's take for
example this small triangle here. What
would the area of this triangle be? You
will say ah that's very easy. It's 15
because you divided that figure into two
triangles. If it's 30 then it would be
15 and 15. Right? Unfortunately, nope.
That's not how it works. We can't
conclude that right away because here,
maybe it's not very clear, but maybe
this part makes it clearer. This
triangle here seems to have a smaller
area than this triangle here. So, we
cannot just divide into two triangles
and say the areas are equal. Noise.
Let's go back to this triangle we were
looking at earlier. The area of a
triangle is base time height / 2. The
height of this triangle would be from
here to here. Right? Do you agree with
me? Now, the height of this triangle
here, look, do you agree that it's the
same as the height of this other
triangle? And this is the most
interesting part. The base of this
triangle here is equal to the base of
this triangle here because those two
segments are equal. So if this triangle
here has the same base as this one on
the left and both triangles have the
same height, what can we conclude? We
can conclude that this area here is
equal to this area here. Look how
beautiful. I will give this a name.
Okay, I will call it A. I could call it
A. I could call it whatever I want. You
can even use another letter. If this
area is called A, then this one will
also be A. Now using this same logic and
I need you to understand this part.
We'll look at these two triangles here.
Notice that both these triangles have
equal bases and the height of this
triangle is also equal to the height of
this one. Right? The heights of the two
are also equal. Therefore, they also
have the same area. Beautiful. Right?
Let us call both of them as B. I think
now you've already caught the mystery
behind this question, right? So these
two triangles with equal bases and equal
heights will have equal areas. I will
now call them C. And of course the same
logic applies here as well. So let us
call them D. Now here comes the real
magic. What can we do next? Let's write
down what we know. I can say that a + d
is 30. Besides that, I know that a + b
is 34. I also know that b + c is 42. And
what about c + d? Actually, I don't know
that. I don't know c + d. But if I am
able to find out c plus d, then my job
is done, right? But hey, we only have
three equations and four unknowns. So at
first it feels like we can't solve it.
So the trick is we don't need all the
unknowns. We only want C plus D and that
we can find smartly. Notice that I have
C here and D here. So it's interesting
to add these two equations. Why? Because
then I will get C + D which I need.
Adding those two equations will be a + d
+ b + c and that's equal to 30 + 42
which is 72. I can reorganize this and
rewrite it as a + b + c + d. And do I
know a + b? Of course I do. It's right
here. A + b is 34. So in place of a + b
I will write 34 here. Now we have 34 + c
+ d is 72. Take 34 right hand side like
this. So c + d = 72 - 34 which is 38.
And that's it. The question is over.
This shaded area here is 38 square
units. Don't forget to share this video.
This question is just too cool. Also,
subscribe to the channel and give it a
like. Always appreciated. So good.
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