# Video: Pack 5 • Paper 3 • Question 17

Pack 5 • Paper 3 • Question 17

03:37

### Video Transcript

𝐴𝐵𝐶𝐷 is a rectangle. 𝑃𝑄𝑅𝑆𝑇 is an irregular pentagon. The perimeter of 𝐴𝐵𝐶𝐷 is equal to the perimeter of 𝑃𝑄𝑅𝑆𝑇. Assuming that measurements are given in centimeters, calculate the area of the rectangle 𝐴𝐵𝐶𝐷.

The perimeter of rectangle 𝐴𝐵𝐶𝐷 is the distance around the outside. This is given by two multiplied by two 𝑥 plus eight plus two multiplied by three 𝑥 minus one as lengths 𝐴𝐵 and 𝐷𝐶 are equal and lengths 𝐴𝐷 and 𝐵𝐶 are also equal. The perimeter of the pentagon 𝑃𝑄𝑅𝑆𝑇 is given by two multiplied by five 𝑥 minus six plus two multiplied by three 𝑥 plus four 𝑥 plus one. The lengths 𝑃𝑄 and 𝑄𝑅 are equal. Also, the lengths 𝑃𝑇 and 𝑅𝑆 are equal. Our final length is the base 𝑆𝑇.

Expanding the first bracket gives us four 𝑥 plus 16. The second bracket on the left-hand side gives us six 𝑥 minus two. Simplifying the right-hand side gives us 10𝑥 minus 12 plus six 𝑥 plus four 𝑥 plus one. Grouping the likes terms on the left-hand side gives us 10𝑥 plus 14 as four 𝑥 plus six 𝑥 is equal to 10𝑥 and 16 minus two is equal to 14. On the right-hand side, we’re left with 20𝑥 minus 11.

In order to balance the equation, we then need to add 11 to both sides of the equation and subtract 10𝑥 from both sides of the equation. This leaves us with 25 is equal to 10𝑥. Dividing both sides of this equation by 10 gives us a value for 𝑥 of 2.5. This means that the length of 𝑥 in both of the shapes is 2.5 centimeters.

The question asked us to calculate the area of the rectangle 𝐴𝐵𝐶𝐷. We do this by multiplying the base by the height. The base of the rectangle can be calculated by substituting 𝑥 equals 2.5 into the expression two 𝑥 plus eight: two multiplied by 2.5 plus eight. Two multiplied by 2.5 is equal to five. Adding eight to this gives us 13. Therefore, the base of the rectangle is 13 centimeters. The height of the rectangle can be calculated by multiplying three by 2.5 and subtracting one. This is equal to 6.5.

Therefore, the area of the rectangle is 13 centimeters multiplied by 6.5 centimeters. This is equal to 84.5 centimeters squared. Therefore, the area of the rectangle is 84.5 centimeters squared.