### Video Transcript

Given that line ππ is parallel to
line πΎπ·, which of the following has the same area as triangle πππΎ? (A) Triangle πΆππ», (B) triangle
πΆππ», (C) triangle π»ππ, (D) quadrilateral ππππ», or (E) quadrilateral
π»ππΎπΆ.

Letβs begin by highlighting the
shape weβre looking at. Triangle πππΎ is highlighted
here. Weβre looking for a shape that has
an equal area. So letβs recall how to find the
area of a triangle. For a triangle whose height is β
units and base is π units, the area of that triangle will be equal to one-half
times the base times the height. Recall that the base of a triangle
doesnβt necessarily need to be at the bottom. That just depends on its
orientation.

So here we can label the base as the
length of line segment ππ. The height is then the
perpendicular distance between line segment ππ and the line that has point πΎ on
it. Weβve already been told that the
line passing through ππ is parallel to the line passing through πΎπ·. This means that the perpendicular
distance between the line ππ and the line πΎπ· can be labeled as β units. It doesnβt matter where on the line
weβre looking. This is really useful because we
now know that any triangle created here will have a height of β units. The triangle πΆππ» would have a
perpendicular height of β units, as would the height of the triangle ππ·π.

Weβve shown so far that triangles
πππΎ, πΆππ», and ππ·π have the same height, β units. Looking closely, we see that the
bases of these triangles have been marked out with a dash mark. This means we can go further and
say that these three triangles have the same base, π units. This means weβve shown that the
area of these three triangles will be the same.

However, we need to go ahead and
consider the other two quadrilaterals that were listed here. ππππ» Iβve highlighted here in
yellow and π»ππΎπΆ in green. These quadrilaterals have one pair
of parallel sides, which means theyβre trapezoids. The area of a trapezoid is equal to
one-half base one plus base two times the height. While these trapezoids do share the
same height as the triangles weβve already mentioned, thatβs the perpendicular
distance between the lines ππ and πΎπ·, in order for one of these trapezoids to
have the same area as triangle πππΎ, we would have to be able to prove that the
two parallel bases are equal in length to the base ππ. And thereβs nothing on this diagram
that gives us enough information or to indicate that this would be the case.

What we can be certain of is that
the area of triangle πππΎ is equal to the other two triangles weβve
considered. This is option (B) triangle
πΆππ». With regard to the other two
triangles in this list, triangle πΆππ» and triangle π»ππΆ, it is true that these
two triangles would have the same height as our three triangles weβve already
considered. However, we have no information
about the bases of these triangles. And therefore, we cannot claim that
they have the same area as triangle πππΎ. From this list, the only triangle
that certainly has the same area as triangle πππΎ is triangle πΆππ».