Determining the Longer Diagonal of a Rhombus with Given Parameters

Consider a rhombus where one of its angles is 60 degrees and its side length is 10 cm. We are tasked with finding the length of the longer diagonal of the rhombus. In this article, we will explore step-by-step the geometric properties and mathematical techniques to determine this length.

Understanding the Properties of a Rhombus

A rhombus is a special type of parallelogram where all four sides are equal in length. One of its key properties is that the diagonals bisect each other at right angles. This means that when the diagonals intersect, they form right-angled triangles.

Determining the Angles of the Rhombus

Given that one angle is 60 degrees, we can deduce the measures of the other angles as follows:

Opposite angles of a rhombus are equal. Therefore, the opposite angle is also 60 degrees. The sum of the angles in any quadrilateral is 360 degrees. The other two angles are thus each 120 degrees (360 - 2*60 - 2*120 120).

Using the Diagonals to Form Right-Angled Triangles

Every rhombus can be divided into four right-angled triangles by its diagonals. Each diagonal bisects the angles of the rhombus, and the diagonals are perpendicular to each other. We can focus on one of these right-angled triangles to find the lengths of the diagonals.

Labeling the Diagonals

Let:

d1 be the length of the shorter diagonal. D be the length of the longer diagonal.

Applying Trigonometric Relationships

Consider the triangle formed by half of the diagonals and one side of the rhombus. We can use the sine function to relate the sides of this triangle:

sin(60°) (d1/2) / 10 Solving for d1, we get:

d1 20 * sin(60°) 20 * (sqrt(3)/2) 10 * sqrt(3) cm

Using the Pythagorean Theorem

By applying the Pythagorean theorem to the same right-angled triangle, we can relate the lengths of the diagonals:

(d1/2)2 (D/2)2 102 Substituting the value of d1 found previously:

(10 * sqrt(3)/2)2 (D/2)2 100 Simplifying further:

(5 * sqrt(3))2 (D/2)2 100 75 (D/2)2 100 (D/2)2 25 D/2 5 D 10 cm

Therefore, the length of the longer diagonal, D, is 10 cm.

Alternative Method Using Law of Cosines

Alternatively, we can use the law of cosines to verify the length of the shorter diagonal:

d2 L2 L2 - 2 * L * L * cos(60°) Substituting the values:

d2 102 102 - 2 * 10 * 10 * cos(60°) d2 100 100 - 100 * (1/2) d2 100 d 10 cm (shorter diagonal)

For the longer diagonal, applying the law of cosines at the other vertex:

D2 102 - 2 * 10 * 10 * cos(120°) D2 100 - 200 * (-1/2) D2 300 D 10 * sqrt(3) cm

Thus, the length of the longer diagonal is 10 * sqrt(3) cm.