Final answer:
To find the diagonals of a rhombus with 10 cm sides and a 60° angle, we use trigonometry and rhombus properties, resulting in one diagonal being 10 cm and the other being 10√3 cm.
Step-by-step explanation:
The student asks about finding the diagonals of a rhombus when given the length of its sides and one of its interior angles. Since a rhombus is a type of parallelogram, the properties of parallelograms are used to solve this problem.
To find the diagonals of a rhombus given the sides are 10 cm long and one angle is 60°, we can use trigonometry and the properties of a rhombus. The diagonals of a rhombus bisect each other at right angles and bisect the angles of the rhombus. We can construct two congruent equilateral triangles by drawing a line from one angle to the opposite angle since all angles in an equilateral triangle are 60°. The longer diagonal of the rhombus can be found by the formula d1 = 2 × side × sin(angle/2), which becomes d1 = 2 × 10 cm × sin(30°). With sin(30°) being 0.5, the longer diagonal is 10 cm.
For the shorter diagonal, we can divide the rhombus into two 30-60-90 right triangles. We can then use the fact that the ratios of the lengths of the sides opposite the 30°, 60°, and 90° angles are 1:√3:2. Here, the side opposite the 30° angle is half of the shorter diagonal, and the side opposite the 60° angle is the length of the rhombus's side. Thus, the shorter diagonal is 2 × side × sin(60°) which becomes d2 = 2 × 10 cm × √3/2, resulting in d2 = 10√3 cm.