– English Lyrics

Lyrics & Translation

[English]

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.

Key Vocabulary

Vocabulary	Meanings
calculate /ˈkælkjəleɪt/ B1	verb - to determine the amount or number of something mathematically
area /ˈɛriə/ A2	noun - the size or extent of a surface or piece of land
region /ˈriːdʒən/ A2	noun - a particular area or part of a country or the world
square /skwɛər/ A1	noun - a plane figure with four equal straight sides and four right angles adjective - having the shape of a square
midpoint /ˈmɪdˌpɔɪnt/ B2	noun - the point or place exactly halfway between two points
quadrilateral /kwɒdrɪˈlætərəl/ B2	noun - a four-sided polygon
triangle /ˈtraɪæŋɡəl/ A1	noun - a plane figure with three straight sides and three angles
base /beɪs/ A2	noun - the bottom or supporting part of something
height /haɪt/ A1	noun - the distance from the bottom to the top of something
equal /ˈiːkwəl/ A1	adjective - being the same in quantity, size, degree, or value
conclude /kənˈkluːd/ B1	verb - to decide or deduce something after consideration
logic /ˈlɒdʒɪk/ B1	noun - a particular way of thinking, especially one that is reasonable and based on good judgment
equation /ɪˈkweɪʒən/ B1	noun - a mathematical statement that two expressions are equal
unknown /ʌnˈnoʊn/ A2	adjective - not known or familiar
smartly /ˈsmɑːrtli/ B2	adverb - in a clever or intelligent way
shaded /ʃeɪdɪd/ A2	adjective - covered or protected from direct light