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# CA Geometry: Pythagorean theorem, area

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We're on problem 36. And it says, what is the area
in square units of the trapezoid shown below? So, when you just look at this
you're like, OK, trapezoid, do I know the formula for the
area of a trapezoid. Then you might get confused
and all that. But you say, well, trapezoid
I can break that up into a rectangle and a triangle. If were to draw a
line right here. Then I've broken up the
trapezoid into a rectangle and a triangle. And if I know the dimensions of
each of those, I know the area of each of them and
then I know the area of the entire thing. So let's see, what's this height
right Here Or this width I should say. Well we're going from
zero to what? x is equal to 8 here. I just went straight down
from x is equal to 8, y is equal to 5. So this dimension is 8. And then we go from x is equal
to 8 to x is equal to 12, how far is that? Well, that's going to be 4. So this is 4 and this is 8. Fair enough. And then how high is
this rectangle? We're going from y is equal
to zero to y is equal to 5, so that's 5. And of course this
is 5 as well. So we're done. We're ready to figure
out the area. The area of the rectangle part
is 8 times 5, that's 40. The area of this triangle
is 5 times 4 times 1/2. If we didn't put that 1/2 we
would be figuring out the area of this rectangle right there. So 5 times 4 is 20
times 1/2 is 10. So the area of both of these
combined is 10 plus 40 is 50. 37. The figure below is a square
with 4 congruent parallelograms in side. This looks interesting. What is the area in square units
of the shaded portion? So the shaded portion is the
whole square minus the area of the parallelogram. So the whole square, that's
easy, it's 12. And the height is 12, but since
you know it's a square we know the width also
has to be 12. So the area of the whole
square is 144. If we know the area of one of
the parallelograms, we know the area of all of the
parallelograms because they are congruent. So let's see if we can figure
out the area of one of the parallelograms. So there is actually a formula
for the area of a parallelogram, it's
actually just the base times the height. And they actually
give us that. But let me show you that they
give us that because it might not be obvious to you. Let me try to draw it. I'll use my line tool. Nope, that's not
the line tool. One side, then go straight
like that, come down like that, good enough. OK, now if I look at just this
parallelgoram, they tell us that the height here is 3. And I know it's the height
because they told me it's a 90 degree angle. And they tell us at
the base is 5. And I'm telling you that the
area of a parallelogram is just the base times the
height is equal to 15. But you shouldn't just
take my word for it. That should make intuitive
sense to you. And the way to think about it
intuitively is imagine if we were to take this part of the
parallelogram, and if we were to move it over here. If we were to cut that off
and move it over here. Then the parallelogram would
look something like this. You'd have the part that
we didn't cut off. And then you move the cut-off
part over here. And now the dimensions, this
base would be 5, and this height would be 3. And the area of this
rectangle is 15. And there's no reason why the
area of this should be any different than that. We just rearranged its parts. So that's why the area of a
parallelogram is just the base times the height. So the area of each of these
parallelograms is 15. So the area of all of
them combined is 15 times 4, which is 60. So 144 minus 60 is 84. And that's choice B. Problem 38. What is the area in square
meters of the trapezoid shown below. So to figure out the area and
we could break it up into these rectangles
and triangles. To figure out the area
of this rectangle we need to know its height. And actually we'll need that to
figure out the area of the triangles as well. So what's this height
right there? Let's see, we know that this
distance is going to be 6. It's a rectangle. If that distance is 6 and both
of these are 5, both of these triangles here are going
to be congruent. Because this length is
equal to this length. This length is equal
to this length. And we also we make this angle
is equal to that angle. But anyway, let me do
it in another color. What's the length of these
two green sides? Let's call it x. Well we know that when you
add x plus 6 plus x it has to equal 12. The whole top part. So you get x plus x is 2x
plus 6 is equal to 12. 2x is equal to 6. x is equal to 3. And you might have been able
to solve that in your head. That if that's 6 and these are
the same, then both of these are going to be 3. And now we can use that
information to figure out this height right there. Because if we just draw this
triangle right there, that's 3, that's 5, this is some
unknown side, a. You might already recognize,
we're going to use the Pythagorean theorem. And this is a very typical
type of right triangle. So might already be able
to guess what a is. But we'll solve for it. So we know that a squared plus
3 squared is equal to the hypotenuse squared, the side
opposite the 90 degree angle. So that's equal to 25. 5 squared is 25. a squared plus 9
is equal to 25. a squared is equal to 16. a is equal to 4. a is equal to 4. And now we're ready to
figure out the area. What's the area of
the rectangle? 6 times 6, it's 24. What's the area of each
of these triangles? 3 times 4 times 1/2. 3 times 4 is 12 times
1/2 is 6. So the area of that
triangle is 6. The area of this
triangle is 6. So 6 plus 24 plus 6 is 36. B. Problem 39. What is the area in square
inches of the triangle below. Interesting. OK, so this is an equilateral
triangle, all the sides are equal. And so we could actually say
that since these two triangles are symmetric. That's equal to that. And this comes to a general
formula for the area of an equilateral triangle. But let's just figure
it all out. So this side is going to be 5. And this side is
going to be 5. If this is 5 and that's 10, what
is this side right here? Let's call it x. Pythagorean theorem. This is the hypotenuse. So x squared plus 5 squared plus
25 is going to be equal to the hypotenuse squared,
it's equal to 100. x squared is equal to
100 minus 25, 75. x is equal to the square
root of 75. 75 is 25 times 3. So that's equal to the square
root of 25 times 3. Which is equal to the square
root of 25 times the square root of 3. Which is equal to
5 roots of 3. And now, what's the area
of just this right triangle right here? This one on the right side. Well its base is 5, its height
is 5 roots of 3. So it's going to be 1/2 times
the base, 5, times the height, 5 roots of 3. And that's what? 1/2 times 5 times 5. So it's 25 root 3 over 2 and
that's just this triangle right there. Well this triangle's going to
have the the exact same area. They are congruent triangles. So the area of the figure
is this times 2. So 2 times that is equal
to just the 25 root 3. And that's choice B. Next problem, problem 40. The perimeter of two squares
are in a ratio of 4:9. What is the ratio between the
areas of the two squares? Let me draw two squares. That's one square. Let me draw another square. That's another square. Let's say that the sides of this
are x and the sides of this one are y. So they say the perimeters of
those two squares are in a ratio of 4:9. So the perimeter of the
first square is 4x. x plus x plus x plus x. So the perimeter of the
first square is 4x. The perimeter of the second
square is 4y. So that's the ratio of the
perimeter of the first square to the perimeter of
the second square. And then that is equal to 4:9. And they say, what is the ratio
between the areas of the two squares? So they want us to figure out
the area of the first square is x squared. Base times height, x times x. And the are of the second
square is y times y. So they want us to figure out
what that is equal to. Well this is x squared
over y squared. This is the same thing
as x over y squared. So if we can figure out what x
over y is equal to, we can just square it and we'll get
x squared over y squared. So let's try to do that. So they gave us this. Well this just simplifies. x over y is equal to 4 over 9. So let's substitute that here. So x squared over y squared is
equal to x over y squared. Which is equal to 4/9 squared. Which is equal to 16 over 81. Or the ratio of the areas of
the two squares is 16:81. Choice D. I think we can fit one more
problem in there. Actually no, I'm over
10 minutes. I'll stop right there. See you in the next video.