To find the lengths of the diagonals of rhombus ABCD, we need to apply the properties of a rhombus.
1. In a rhombus, all sides are congruent. Since AB = 10, all sides of the rhombus are also 10 units in length.
2. In a rhombus, opposite angles are congruent. Thus, ∠A = ∠C, and ∠B = ∠D.
3. The diagonals of a rhombus bisect each other at right angles. This means that the diagonals intersect at a 90-degree angle.
Now, let's proceed with finding the lengths of the diagonals.
Since ∠A = 60°, we know that ∠C (opposite angle) is also 60°.
Using the properties of a rhombus, we can apply the Law of Cosines to find the lengths of the diagonals.
Let AC represent the length of one diagonal, and BD represent the length of the other diagonal.
Using the Law of Cosines for triangle ABC, we can express AC in terms of the known sides:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(∠ABC)
Since the rhombus has congruent angles, we have:
AC^2 = 10^2 + 10^2 - 2 * 10 * 10 * cos(60°)
AC^2 = 100 + 100 - 200 * 0.5
AC^2 = 200 - 100
AC^2 = 100
Taking the square root of both sides, we find:
AC = sqrt(100)
AC = 10
So, one diagonal, AC, has a length of 10 units.
Since the diagonals of a rhombus bisect each other at right angles, we can use the Pythagorean Theorem to find the length of the other diagonal, BD.
BD^2 = AB^2 + AD^2
where AD is half of AC.
AD = AC / 2 = 10 / 2 = 5
BD^2 = 10^2 + 5^2
BD^2 = 100 + 25
BD^2 = 125
Taking the square root of both sides, we find:
BD = sqrt(125)
BD = 11.18 (approx.)
So, the lengths of the diagonals of rhombus ABCD are approximately 10 units and 11.18 units, respectively.