In a rhombus, the diagonals bisect each other at right angles. This means each diagonal is divided into two equal parts at the intersection point.
Given:
- The length of diagonal QS is 42. Therefore, the length of QU (half of QS) is: \[ QU = \frac{QS}{2} = \frac{42}{2} = 21. \]
- The length of diagonal RT is 40. Therefore, the length of RU (half of RT) is: \[ RU = \frac{RT}{2} = \frac{40}{2} = 20. \]
Now, to find the lengths of the sides of the rhombus, we can apply the Pythagorean theorem. The sides of the rhombus (e.g., ST) can be found using the right triangle formed by points S, T, and U (where ST is one of the sides of the rhombus).
Using the Pythagorean theorem: \[ ST^2 = SU^2 + UT^2. \] Where SU = QU = 21 and UT = RU = 20.
Calculating for ST: \[ ST^2 = 21^2 + 20^2 = 441 + 400 = 841. \] Thus: \[ ST = \sqrt{841} = 29. \]
Therefore, the length of ST is 29.