Given that the area of the
parallelogram 𝐴𝐵𝐶𝐷 is 24 square centimeters and the area of the rectangle
𝑋𝐵𝑌𝐷 is 12 square centimeters, find the perimeter of the rectangle 𝑋𝐵𝑌𝐷.
We are given on the diagram that
the length of 𝐴𝑋 is three centimeters. We know that the parallelogram
𝐴𝐵𝐶𝐷 has area 24 square centimeters. The area of the rectangle 𝑋𝐵𝑌𝐷
is 12 square centimeters. And we need to calculate the
perimeter of this shape. The triangles 𝑋𝐴𝐷 and 𝑌𝐶𝐵 are
congruent. This means that they have the same
area. This means that we can calculate
the area of triangle 𝑋𝐴𝐷 by subtracting 12 from 24 and then dividing by two.
Subtracting the area of the
rectangle from the area of the parallelogram will give the area of both
triangles. As the triangles are congruent, we
then need to divide by two. 24 minus 12 divided by two is equal
to six. The area of triangle 𝑋𝐴𝐷 is six
We know that to calculate the area
of any triangle, we multiply the base by the height and then divide by two. We already know that the base of
this triangle is three centimeters. This means that six is equal to
three multiplied by ℎ divided by two. Multiplying both sides of this
equation by two gives us 12 is equal to three ℎ. We can then divide both sides by
three, giving us ℎ is equal to four. The height of the triangle 𝑋𝐷 is
equal to four centimeters.
We know the area of any rectangle
is equal to its base multiplied by its height. We know the height of the rectangle
is four centimeters and its area is 12 square centimeters. Substituting in these values gives
us 12 is equal to 𝑏 multiplied by four. Dividing both sides of this
equation by four gives us 𝑏 is equal to three. The base or length of the rectangle
𝑋𝐵 is equal to three centimeters.
We now have a rectangle 𝑋𝐵𝑌𝐷
with dimensions four centimeters and three centimeters. The opposite sides of a rectangle
are equal in length, and the perimeter is the distance around the outside. We can therefore calculate the
perimeter by adding two threes and two fours. This is equal to 14. The perimeter of rectangle 𝑋𝐵𝑌𝐷
is 14 centimeters.