Question Video: Finding the Area of a Composite Figure Inscribed between a Square and a Rectangle

Video Transcript

Given that 𝐴𝐵𝐶𝐷 is a rectangle, 𝑊𝑋𝑌𝑍 is a square, 𝐴𝐵 equals six centimeters, and 𝐵𝐶 equals three centimeters, calculate the area of the shaded part.

We can calculate the area of the shaded region by subtracting the area of the square from the area of the rectangle. The area of any rectangle is equal to its length multiplied by its width. We are told in this question that the length 𝐴𝐵 is equal to six centimeters, the width 𝐵𝐶 is equal to three centimeters. This means that the area of the rectangle is equal to six centimeters multiplied by three centimeters. Six multiplied by three is equal to 18. Therefore, the area of the rectangle is 18 centimeters squared.

As the width of the rectangle is equal to three centimeters, the length of the diagonal of the Square from 𝑋 to 𝑍 must also be three centimeters. The length 𝑊𝑌 is also equal to three centimeters, as the diagonals of a square are equal in length. The diagonals of the square bisect at the center of the square. This means that the length from the center 𝑜 to 𝑌 is 1.5 centimeters, as a half of three is 1.5.

We have split a square into two equal-sized triangles. The area of any triangle can be calculated by multiplying its base by the perpendicular height and dividing this answer by two. This means that the area of triangle 𝑋𝑌𝑍 is equal to three multiplied by 1.5 divided by two. Three multiplied by 1.5 is equal to 4.5. Dividing this by two gives us 2.25 centimeters squared. As the square is made up of two identical triangles 𝑋𝑌𝑍 and 𝑋𝑊𝑍, we need to multiply 2.25 by two. The area of the square is 4.5 centimeter squared. We can now calculate the area of the shaded part by subtracting 4.5 from 18. This is equal to 13.5. The area of the shaded part on the diagram is 13.5 centimeters squared or 13.5 square centimeters.