Video Transcript
Olivia is a set designer. She wants to use two wooden pieces,
a square 𝐴𝐵𝐶𝐷 and a right triangle 𝐴𝐷𝐸, which are attached together along
𝐴𝐷 by hinges. She wants to place them on the
floor, as shown in the figure, and make sure that the two pieces form a right
angle. If this is the case, then any line
in the triangle is perpendicular to the line segment 𝐴𝐵, and any line in the
square is perpendicular to the line segment 𝐴𝐸. She has no tool with her except a
measuring tape. Part one, she measured the length
of line segment 𝐸𝐵 and found it to be exactly 42 centimeters. Is the angle between the square and
the triangle a right angle? Part two, given that the line
segment 𝐵𝐶 is perpendicular to the floor, Olivia could have measured the length of
the line segment 𝐸𝐶 to determine whether the two wooden pieces formed a right
angle. What would the length of the line
segment 𝐸𝐶 have been if it were a right angle? Round your answer to the nearest
tenth.
Now, if we take a look at the first
part of the question, what we’re trying to do is decide whether the triangle formed
out of 𝐸𝐴𝐵 is going to be a right triangle. So what we’ve done is drawn a
sketch. We’ve got the line segment 𝐸𝐵 as
42 centimeters. Then, we’ve got 28.3 centimeters
and 30 centimeters as the other two sides. So what we’re trying to do is
determine whether the angle between the square and the triangle is a right
angle. So it’s going to be the angle at
𝐴.
And to do this, what we’re going to
do is use the converse of the Pythagorean theorem. And what the Pythagorean theorem
states is that 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎, 𝑏, and 𝑐
are the sides of a right triangle, with 𝑐 being the hypotenuse. So what it says is that the square
of the two shorter sides added together equals the square of the longest side, our
hypotenuse. Well, I already stated that we
would use the converse of this to help solve this problem because if in fact we do
have a right triangle 𝐸𝐴𝐵, then this should satisfy the Pythagorean theorem. So what we can do is see if this is
in fact the case.
Now, to see whether this is the
case, the first thing we will do is square the two shorter sides and add them
together, so 28.3 squared plus 30 squared. Well, this is going to be
1700.89. Okay, great. So now what we can do is square the
longer side. So this means that we can square
42. Well, when we square 42, what we
get is 1764. So now what we can do is compare
our 𝑎 squared plus 𝑏 squared and our 𝑐 squared. And when we do that, we can see
they are not in fact the same because our 𝑎 squared plus 𝑏 squared, so the squares
of the two shorter sides, is 1700.89, whereas the square of the longest side is
1764. So therefore, what we can say is
that in answer to the question “Is the angle between the square and triangle a right
angle?,” our answer would be no.
Now we’re about to move on to the
second part of the question, which tells us that given that the line segment 𝐵𝐶 is
perpendicular to the floor, Olivia could have measured the length of the line
segment 𝐸𝐶 to determine whether two pieces formed a right angle. And what we’re trying to do is see
what the length of the line segment 𝐸𝐶 would have been if it were a right
angle. Well, what we need to do to help us
with this is in fact find out what the length of 𝐸𝐵 would have been if in fact the
triangle that we talked about, 𝐸𝐴𝐵, was a right angle. And we can do that by taking the
square root of our 1700.89. And if we take the square root of
1700.89, we get 41.241. So this would be the length of our
line segment 𝐸𝐵.
Now, what we’re going to do is
clear some space so we can answer the second part of the problem.
So, what we’ve done is drawn a
sketch as before. In fact, I’ve actually drawn a
sketch on the original diagram just to show where our right triangle comes from. We’ve been told to assume that it
is a right triangle. So we have the triangle 𝐸𝐵𝐶,
where the side 𝐵𝐶 is 28.3 centimeters. And that’s because we know that
because it’s one of the sides of our yellow square. And we know that 𝐸𝐵 is 41.241
centimeters because we calculated that just now. Now, we’re told to assume, like we
said, that it’s a right triangle. So therefore, what we can do is use
the Pythagorean theorem to calculate 𝐸𝐶.
As we already stated in the first
part of the question, the Pythagorean theorem is 𝑎 squared plus 𝑏 squared equals
𝑐 squared, where 𝑐 is the hypotenuse. So therefore, we can say that 𝐸𝐶
squared is gonna be equal to 28.3 squared plus 41.241 continued squared. So this means that 𝐸𝐶 squared is
gonna be equal to 800.89 plus 1700.89. And we know that that’s the case
without actually calculating it because if we look back, we took the square root of
1700.89 to get our 41.241 continued. So if we add these together, we get
2501.78.
So now if we want to find 𝐸𝐶,
what we do is take the square root of both sides of the equation. And when we do that, we’re gonna
get 𝐸𝐶 is equal to 50.017 continued. And then checking the question to
see how it wants its answer left, it wants the answer rounded to the nearest
tenth. So therefore, what we can say is
that the length of the line segment 𝐸𝐶 would have been 50.0 centimeters to the
nearest tenth if in fact two wooden pieces formed a right angle.
