### Video Transcript

π΄π΅πΆ is an isosceles triangle
with a right angle at π΅. The points π·, πΈ, and πΉ are the
midpoints of the line segments π΄πΆ, π΄π΅, and π΅πΆ, respectively. There are three parts to this
question. Part one asks us, which of the
given planes is an orthonormal coordinate plane? Part two asks us, which of the
following planes is an orthogonal but not an orthonormal coordinate plane? And part three asks us, which of
the following planes is an oblique coordinate plane?

So letβs look at part one. Which of the following planes is an
orthonormal coordinate plane? Is it option (A) π΄; πΈ, π·? Option (B) π΅; πΆ, πΈ. Option (C) plane π΅; πΉ, πΈ. Option (D) π΄; π΅, πΆ. Or option (E) plane πΆ; π΄, π΅.

To answer this, we recall that when
defining a coordinate plane π; πΌ, π½, the first point given is the origin of the
coordinate plane. The line from the origin through
the second point, that is, the line ππΌ forms the π₯-axis and the line from the
origin through the third point, that is, the line ππ½ forms the π¦-axis. In part one, weβre looking for an
orthonormal coordinate plane, thatβs one where the two axes are perpendicular, and
the length from the origin to the points πΌ and π½. That is, the unit lengths are
equal. So letβs begin by going through
each of the given options to see which satisfy the perpendicularity criterion.

In option (A), our origin is the
point π΄. The axes are the lines π΄πΈ and
π΄π·. But since our triangle π΄π΅πΆ is
right at angle π΅, then angle π·π΄π΅ cannot be a right angle. It must be less than 90 degrees,
and therefore our axes cannot be perpendicular. And so we can eliminate option
(A). In option (B), π΅ is the origin,
and the axes are π΅πΆ and π΅πΈ. And since our triangle π΄π΅πΆ has a
right angle at π΅, then the two axes π΅πΆ and π΅πΈ are indeed perpendicular. So the first criteria, thatβs the
perpendicularity criteria, is satisfied for option (B). Since our triangle π΄π΅πΆ is
isosceles however, the unit lengths π΅πΆ and π΅πΈ are not equal. The side lengths π΅π΄ and π΅πΆ are
equal. However, π΅πΈ is only half of the
length of π΅π΄. And that is half of the length of
π΅πΆ. And since our unit lengths are not
equal, we can eliminate option (B).

Now considering option (C), again
we have π΅ as the origin. Our axes in this case are π΅πΉ and
π΅πΈ. And we see then since π΅ is the
origin, our axes are perpendicular. And so our first criteria is
satisfied. Now, since πΉ and πΈ are the
midpoints of π΅πΆ and π΅π΄, respectively, and the triangle is isosceles, we know
that side lengths π΅π΄ and π΅πΆ are the same. We have π΅πΈ is equal to one over
two π΅π΄. π΅πΉ is a half π΅πΆ, and these are
equal. And so for option (C), our second
criteria is also satisfied. The unit lengths are equal. And so the coordinate plane defined
in option (C) is an orthonormal coordinate plane.

If we look at our remaining
options, thatβs (D) and (E), in option (D), the origin is at the point π΄. So as with option (A), we can
discount this, since the angle that π΄ is not a right angle and hence the axes are
not perpendicular. That is, the axes π΄π΅ and π΄πΆ are
not perpendicular. And finally, in option (E), we have
πΆ as the origin. The axes are πΆπ΄ and πΆπ΅. And so the angle between them
cannot be 90 degrees. The axes are not perpendicular. Hence, we can discount option
(E). And hence, the answer to part one
of the question, which of the planes is orthonormal?, is option (C). Thatβs the plane π΅; πΉ, πΈ.

Now moving on to part two, which of
the following planes is an orthogonal but not an orthonormal coordinate plane? Option (A) plane π΅; πΉ, πΈ. Option (B) plane π΅; πΆ, π΄. Option (C) plane π·; π΅, πΆ. Option (D) plane π΄; π΅, πΆ. Or option (E) plane π΅; πΉ, π΄.

Now in an orthogonal plane, the
axes are perpendicular. But since the plane weβre looking
for is not orthonormal, then our unit lengths will not be equal. So letβs consider our five
options. Options (A), (B), and (E) have π΅
as the origin. Option (A) has the axes π΅πΉ and
π΅πΈ. And these are indeed perpendicular,
so the first criteria is satisfied. Option (B) has axes π΅πΆ and
π΅π΄. And these are perpendicular, so
option (B) satisfies the first criteria. And option (E) has axes π΅πΉ and
π΅π΄. These also are perpendicular, so
our first condition of perpendicularity is satisfied for option (E) also.

Now option (C) has π· as the
origin. And since our triangle π΄π΅πΆ is
isosceles, the point π· is the perpendicular bisector of π΄πΆ. And this means that our axes π·π΅
and π·πΆ are indeed perpendicular. So the axes for option (C) are
perpendicular. Now considering option (π·), we see
that the origin is the point π΄. With axes π΄π΅ and π΄πΆ, we know
that the angle between them cannot be 90 degrees. And so in this case, we can
eliminate option (D).

So we still have options (A), (B),
(C), and (E) to consider. We know that we donβt want our unit
length to be the same so that our coordinate plane is not an orthonormal coordinate
plane. So letβs look at these four
remaining options. In option (A), the unit lengths are
π΅πΉ and π΅πΈ. Now π΅πΉ is the midpoint of π΅πΆ,
so π΅πΉ is one-half π΅πΆ. And π΅πΈ is the midpoint of π΅π΄,
so π΅πΈ is one-half of π΅π΄. But since our triangle π΄π΅πΆ is an
isosceles triangle, π΅π΄ is equal to π΅πΆ. So one over two π΅πΆ is one over
two π΅π΄. And this means that π΅πΉ is indeed
equal to π΅πΈ. This means that the unit lengths
for option (A) are equal. So we can eliminate option (A). In fact, by the same logic, we can
eliminate option (B). Since sides π΅π΄ and π΅πΆ are the
equal sides of an isosceles triangle, so option (B) does not satisfy our second
criteria. And we can eliminate option
(B).

Next, looking at option (C), our
unit lengths are the lengths π·π΅ and π·πΆ. And if we consider our triangles,
triangle π΄π΅πΆ is isosceles, so angle π΅π΄πΆ is equal to angle π΅πΆπ΄, and thatβs
45 degrees. And since the line segment π·π΅
bisects the angle π΄π΅πΆ, which is 90 degrees, we have angle πΆπ΅π· is 90 over two,
and thatβs 45 degrees. So now, weβre looking at triangle
π΅πΆπ·. And this is also an isosceles
triangle. And so the side lengths π·π΅ and
π·πΆ are in fact equal. This means that the unit lengths
for option (C) are equal. Hence, we can eliminate option (C),
since option (C) represents an orthonormal coordinate plane.

So now finally considering option
(E), we have π΅ as our origin and axes π΅πΉ and π΅π΄. And we know already that π΅πΉ is
actually one-half of π΅π΄, since π΅π΄ is the same as π΅πΆ and πΉ is the midpoint of
π΅πΆ. So for option (E), our unit lengths
are not the same and the plane represented by (E) is not orthonormal, but it is
orthogonal. So the answer to part two is option
(E), the plane π΅; πΉ, π΄.

So now letβs look at part
three. This asks us, which of the
following planes is an oblique coordinate plane? Option (A) plane π·; π΅, πΆ. Option (B) plane π΅; πΆ, π·. Option (C) plane π΅; πΆ, π΄. Option (D) plane π·; π΅, π΄. Or option (E) plane πΈ; π΅, π·.

So weβre looking for an oblique
coordinate plane that is a coordinate plane whose axes are not perpendicular. We see option (A) has its origin at
the point π·. Its axes are π·π΅ and π·πΆ. And weβve seen already that these
are actually perpendicular. So we can eliminate option (A). For option (B) on the other hand,
the origin is at π΅ and the axes are π΅πΆ and π΅π·. And weβve seen already that the
angle between these two is not 90 degrees. Hence, the axes for option (B) are
not perpendicular. So (B) does represent an oblique
coordinate plane.

Option (C) has its origin at π΅
with axes π΅πΆ and π΅π΄. And these are perpendicular, so we
can eliminate option (C). Option (D) has its origin at π· and
has axes π·π΅ and π·π΄, which are perpendicular. So we can eliminate option (D). And finally, option (E) has its
origin at point πΈ and axes πΈπ΅ and πΈπ·. And since these axes are
perpendicular, we can eliminate option (E). And of our options, only plane (B)
is an oblique coordinate plane.

Our answer to part one is option
(C), plane π΅; πΉ, πΈ. Our answer for part two is option
(E); thatβs plane π΅; πΉ, π΄. And our answer to part three is
option (B), and thatβs plane π΅; πΆ, π·.