Question

in rhombus pqrs, m∠qrt = 37°, rs = 10 and qt = 6. find the length of \\(\overline{st}\\). note: the diagram is not drawn to scale.

Explanation:

Step1: Recall rhombus properties

In a rhombus, diagonals bisect each other at right angles? Wait, no, diagonals of a rhombus bisect each other and bisect the angles, and the sides are equal. Also, the diagonals are perpendicular? Wait, actually, in a rhombus, the diagonals are perpendicular bisectors of each other. Wait, but in the diagram, we have triangle QRT. Let's see, RS is a side of the rhombus, so RS = RQ = 10 (since all sides of a rhombus are equal). QT is 6, and we need to find ST. Wait, in a rhombus, the diagonals bisect each other, so ST should be equal to QT? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the diagonals of a rhombus bisect each other, so if SQ and PR are diagonals intersecting at T, then ST = TQ. But wait, QT is 6, so ST should be 6? But that seems too easy. Wait, maybe the diagram is different. Wait, no, in a rhombus, the diagonals bisect each other, so the length of ST is equal to the length of QT. Because T is the midpoint of SQ. So if QT is 6, then ST is also 6.

Wait, let's confirm. In a rhombus, the diagonals bisect each other. So diagonal SQ is split into two equal parts by T, so ST = TQ. Given QT = 6, so ST = 6.

Step2: Confirm with rhombus properties

All sides of a rhombus are equal, and diagonals bisect each other. So T is the midpoint of SQ, hence ST = QT = 6.

Answer:

6