Final answer:
In a rhombus, all sides are equal, and the diagonals bisect each other at right angles. Applying the Pythagorean theorem to the diagonals gives us the length of one side. Thus, the length of side DC in the rhombus is approximately 10.97 units.
Step-by-step explanation:
The question asks us to solve for the length of side DC in a rhombus ABCD, where AE = 8 and EB = 15. In a rhombus, all sides are of equal length, so to solve for the length of DC, we can utilize the properties of a rhombus. Since AE and EB are parts of a diagonal that has been split into two segments, we can assume this diagonal is bisected at point E, making AE equal to EC, as the diagonals of a rhombus bisect each other at right angles. Therefore, AE = EC = 8 units. Since the diagonals of a rhombus are perpendicular, we have two right triangles AED and BEC where AD and BC are the hypotenuses. Given AE = EC = 8 and EB = 15, we can use the Pythagorean theorem to find AD and BC (which are of the same length because ABCD is a rhombus).
Using the Pythagorean theorem for triangle AED or BEC:
- AD² = AE² + ED²
- ED is half of DB, since E is the midpoint of DB. Therefore, ED = DB / 2 = EB / 2 = 15 / 2 = 7.5 units.
- AD² = 8² + 7.5²
- AD² = 64 + 56.25
- AD² = 120.25
- AD = √120.25
- AD = 10.97 units (approximately)
Since AD = BC = CD = AB in a rhombus, the length of side DC is also approximately 10.97 units.