### Video Transcript

Given that π΄π΅πΆπ· is a rectangle, where πΈπΆ equals six π₯ minus seven and π΄πΈ equals two π₯ plus five, find π·πΈ.

We know that the two diagonals of a rectangle are equal in length and meet at the center. This means that the length π·π΅ is equal to the length π΄πΆ. As they meet at the center, the distance from each corner or vertex of the rectangle to the center must be the same. The lengths π΄πΈ, π΅πΈ, πΆπΈ, and π·πΈ are all equal.

We are told in the question that the length πΈπΆ, or πΆπΈ, is equal to six π₯ minus seven and the length π΄πΈ is equal to two π₯ plus five. These two expressions must, therefore, be equal to each other. We can solve this equation using the balancing method, firstly by subtracting two π₯ and adding seven to both sides.

On the left-hand side, six π₯ minus two π₯ is equal to four π₯, and the sevens cancel. On the right-hand side, the two π₯βs cancel, and five plus seven is equal to 12. Our final step to solve the equation is to divide both sides by four. This gives us π₯ is equal to three. Substituting this value into our expression for π΄πΈ gives us two multiplied by three plus five. This is equal to 11. So, the length of π΄πΈ is 11 units. As the length of π·πΈ is the same as the length of π΄πΈ, then π·πΈ is equal to 11 units.