Question Video: Finding the Perimeter of an Inscribed Parallelogram given Its Area and the Area of the Parallelogram It Is Inscribed In Mathematics

Video Transcript

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 square centimeters.

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.