Given that 𝐴𝐵𝐶𝐷 is a rectangle, where 𝐸𝐶 equals six 𝑥 minus seven and 𝐴𝐸 equals two 𝑥 plus five, find 𝐷𝐸.
We know that the two diagonals of a rectangle are equal in length and meet at the center. This means that the length 𝐷𝐵 is equal to the length 𝐴𝐶. 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 𝐴𝐸, 𝐵𝐸, 𝐶𝐸, and 𝐷𝐸 are all equal.
We are told in the question that the length 𝐸𝐶, or 𝐶𝐸, is equal to six 𝑥 minus seven and the length 𝐴𝐸 is equal to two 𝑥 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 𝑥 and adding seven to both sides.
On the left-hand side, six 𝑥 minus two 𝑥 is equal to four 𝑥, and the sevens cancel. On the right-hand side, the two 𝑥’s 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 𝑥 is equal to three. Substituting this value into our expression for 𝐴𝐸 gives us two multiplied by three plus five. This is equal to 11. So, the length of 𝐴𝐸 is 11 units. As the length of 𝐷𝐸 is the same as the length of 𝐴𝐸, then 𝐷𝐸 is equal to 11 units.