Finding Angles of a Rhombus When One Diagonal Equals Its Side

When one of the diagonals of a rhombus is equal to its side length, the geometric properties of the rhombus can be used to determine its angles. This article will walk you through the mathematical derivation to find these angles, illustrating how knowledge of right triangles and the Pythagorean Theorem can solve the problem.

Properties of a Rhombus

A rhombus is a special type of parallelogram where all sides are equal in length. Additionally, the diagonals of a rhombus bisect each other at right angles (90 degrees), and each diagonal divides the rhombus into two congruent triangles.

Given Conditions

Let the length of each side of the rhombus be s. One diagonal d_1 is equal to the side length s.

Steps to Find the Angles

Let the diagonals be d_1 and d_2.

d_1 s. d_2 x. Since the diagonals bisect each other at right angles, each half of the diagonals forms two right triangles. The half-lengths of the diagonals are frac{d_1}{2} frac{s}{2} and frac{d_2}{2} frac{x}{2}. Applying the Pythagorean Theorem to one of the right triangles:

s^2 left(frac{s}{2}right)^2 left(frac{x}{2}right)^2

s^2 frac{s^2}{4} frac{x^2}{4}

Multiplying through by 4 to eliminate the fractions:

4s^2 s^2 x^2

Rearranging gives:

3s^2 x^2 quad Rightarrow quad x ssqrt{3}

Finding the Angles

The angles of the rhombus can be determined using the tangent function in the right triangles. For one of the angles theta:

tan{theta} frac{frac{d_2}{2}}{frac{d_1}{2}} frac{frac{ssqrt{3}}{2}}{frac{s}{2}} sqrt{3}

This implies:

theta 60^circ

Since opposite angles in a rhombus are equal and adjacent angles are supplementary, the angles of the rhombus are 60^circ and 120^circ (360 degrees in total).

Therefore, the angles of the rhombus are 60^circ and 120^circ.

The Rhombus as Two Equilateral Triangles

When a side and one diagonal of the rhombus are equal, the diagonal divides the rhombus into two equilateral triangles. This is because each of these triangles has all sides of equal length, and each angle of an equilateral triangle is 60^circ. Thus, the angles of the rhombus are indeed 60^circ and 120^circ.