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Rhombus: Properties and Shape

Sides, Angles and Diagonals

A rhombus is a type of parallelogram, and what distinguishes its shape is that all four of its sides are congruent. There are several formulas for the rhombus that have to do with its
  • Sides (click for more detail)
    • all 4 sides are congruent
  • Angles (click for more detail)
    • diagonals bisect vertex angles
  • Diagonals (click for more detail)
    • diagonals are pependicular
  • Area (click for more detail)
A square is a rhombus and a rectangle. In other words, if each angle of a rhombus is 90°, then it's a square.
Probalby the most famous rhombus out there is the baseball diamond. The distance between each base is the same, making the shape a rhombus!
rhombus as baseball diamond

Rhombus Sides

All sides of a Rhombus are congruent.

AB congruent BC congruent CD congruent AD
Rhombus Picture If side WX = 22, what is WZ?
Side WZ
Sides of A Rhombus

If side MN of rhombus LMNO is
X + 5 and side LM is 2x − 9, what must be the value of x?
Answer
What must be the value of x if side BA = 5x-11
and side AD = 6x-18?
Answer
Since this shape is a rhombus you can set any of its sides equal to each other.
Properties of rhombus

Diagonals of a Rhombus

Diagonals of Rhombus are Perpendicular Diagonals are perpendicular.
  • AOD = 90°
  • AOB = 90°
  • BOC = 90°
  • COD = 90°
Is the four-sided shape below, MNOP, a rhombus? If not, classify the shape.
Answer
Rhombus and Parallelograms Comparison

Anngles of a Rhombus

The diagonals bisect the vertex angles of a rhombus.

A proof of this property of the diagonals
Rhombus Angles
Rhombus Angles Bisect

What is the measure of the following angles in rhombus ABCD?
ACD
ABD
Answer
A generalization about the angles of a rhombus
You can think of a rhombus as four triangles that are created by the diagonals such as . What is true about the outside angles in each triangle? An examples of outside angles are
Answer
What is the value of x if angle BCA = 3x -2 and angle ACD = 12 + x?
Answer
Rhombus
What is the value of x, given the angle measurements below?
Answer
Rhombus Angles Practice Problem

Area of Rhombus


Formula area of rhombus The formula for area of A Rhombus
Area = ½(diagonal1 × diagonal2)
What is the area of HIJK?
Answer

Rhombus Area
Putting It All Together
STAR is a rhombus. The measure of diagonals SA is 24 and the measure of TR is 10, what is the perimeter of this rhombus?
Ask yourself: What is true about the angles formed by the diagonals of a rhombus?
The angles, are perpendicular!
Now, what is true about the diagonals of all parallelograms?
The diagonals of parallelograms bisect each other.
Therefore,
ZA = 12
ZT = 5
What kind of triangle is triangleZTA?
A Right Triangle!
So you can use the pythagorean theorem to find the measure of side TA
Now, that you know the length of TA? How can you use the fact that the sides of a rhombus are congruent to finish this problem?
Since, all 4 sides must be 13. The perimeter = 13 + 13 + 13 +13 = 52

Rhombus Diagonals