Video Transcript
Given that the area of triangle
𝑌𝐴𝐵 is equal to 568 square centimeters, find the area of triangle 𝑋𝐶𝐷.
In this question, we are given the
area of a triangle and a figure and asked to determine the area of another
triangle. To answer this question, we can
begin by highlighting the triangle of known area and the triangle whose area we want
to find on the diagram. At first, it may seem difficult to
compare the areas of these two triangles directly, since there is little information
to go on. Instead, let’s look at the
information we are given in the figure.
We are given no congruent line
segments in the figure. We are only given three parallel
lines. However, we can recall a theorem
for the equality of areas of triangles between parallel lines. We can recall that if two triangles
share a base and the vertices that are opposite the base lie on a straight line that
is parallel to the base, then the triangles must have the same areas. This is because the perpendicular
distance between parallel lines remains constant. So, the perpendicular heights of
the triangles will be equal. We can apply this result to several
triangles in the figure.
First, we note that triangles
𝑌𝑋𝐷 and 𝑌𝑋𝐴 share the line segment 𝑌𝑋 as a base and the vertices opposite
the base lie on a line parallel to the base. Therefore, we can apply this
property to conclude that the area of triangle 𝑌𝑋𝐷 is equal to the area of
triangle 𝑌𝑋𝐴. We can also apply this property to
the triangles on the bottom of the figure. We can see that triangles 𝑌𝑋𝐶
and 𝑌𝑋𝐵 share line segment 𝑌𝑋 as a base and their vertices opposite the base
lie on a line parallel to the base. Hence, the area of triangle 𝑌𝑋𝐶
is equal to the area of triangle 𝑌𝑋𝐵.
We can then note that each of these
pairs of triangles combine to make the two larger triangles that we are interested
in. In particular, the area of triangle
𝑌𝑋𝐷 added to the area of triangle 𝑌𝑋𝐶 gives us the area of triangle
𝑋𝐶𝐷. Similarly, the sum of the areas of
triangles 𝑌𝑋𝐴 and 𝑌𝑋𝐵 is equal to the area of triangle 𝑌𝐴𝐵. Since these triangles are made up
of two triangles of the same area with no overlap, they must have the same area. Hence, the area of triangle 𝑋𝐶𝐷
is also 568 square centimeters.
