Diagonals of a Rhombus Practice Problems and Solutions

Master rhombus diagonal properties with step-by-step practice problems. Learn perpendicular diagonals, angle bisection, and area calculations through interactive exercises.

📚Master Rhombus Diagonal Properties Through Practice
  • Apply the three fundamental properties of rhombus diagonals in problem-solving
  • Calculate rhombus area using the diagonal multiplication formula
  • Prove that diagonals bisect angles and form congruent triangles
  • Solve for unknown diagonal lengths using perpendicular intersection properties
  • Identify when diagonals create 90-degree angles at intersection points
  • Work with midpoint properties where diagonals intersect each other

Understanding Diagonals of a Rhombus

Complete explanation with examples

The diagonals of a rhombus have 3 properties that we can use without having to prove them:

The diagonals of a rhombus have 2 properties that we must prove to use them:

Other properties:

  • The lengths of the diagonals of a rhombus are not equal. except in a square, where the diagonals are equal."

The product of the diagonals divided by 2 is equal to the area of the rhombus:
product of the diagonals2=area of rhombus\frac{product~of~the~diagonals}{2}=area~of~rhombus

Diagonals of a rhombus

A - Diagonals of a rhombus

Detailed explanation

Practice Diagonals of a Rhombus

Test your knowledge with 4 quizzes

Given the rhombus:

BBBAAACCCDDD130

How much is it worth? \( ∢B \)?

Examples with solutions for Diagonals of a Rhombus

Step-by-step solutions included
Exercise #1

Look at the following rhombus:

Are the diagonals of the rhombus perpendicular to each other?

Step-by-Step Solution

The diagonals of the rhombus are indeed perpendicular to each other (property of a rhombus)

Therefore, the correct answer is answer A.

Answer:

Yes.

Exercise #2

Look at the following rhombus:

Are the diagonals of the rhombus parallel?

Step-by-Step Solution

The diagonals of the rhombus intersect at their point of intersection, and therefore are not parallel

Answer:

No.

Exercise #3

Look at the rhombus below:

Are the diagonals of the rhombuses bisectors?

Step-by-Step Solution

To solve the problem, let's review a fundamental property of rhombuses:

  • In a rhombus, the diagonals have a special property: they intersect each other at right angles (90 degrees) and bisect each other. This means each diagonal cuts the other into two equal halves.

Why is this the case? Consider the fact that a rhombus is a type of parallelogram with all sides of equal length. Therefore, each diagonal acts as a line of symmetry, dividing the rhombus into two congruent triangles. This symmetry ensures that the diagonals not only intersect at right angles but also bisect each other.

In summary, given that the shape in question is a rhombus, we can confidently state that the diagonals do bisect each other.

Therefore, the answer to the problem is Yes.

Answer:

Yes

Exercise #4

Do the diagonals of the rhombus above intersect each other?

Step-by-Step Solution

In a rhombus, all sides are equal, and therefore it is a type of parallelogram. It follows that its diagonals indeed intersect each other (this is one of the properties of a parallelogram).

Therefore, the correct answer is answer A.

Answer:

Yes

Exercise #5

Observe the rhombus below:

Determine whether the diagonals of the rhombus form 4 congruent triangles?

Step-by-Step Solution

First, let's mark the vertices of the rhombus with the letters ABCD, then proceed to draw the diagonals AC and BD, and finally mark their intersection point with the letter E:

AAABBBCCCDDDEEE

Now let's examine the following properties:

a. The rhombus is a type of parallelogram, therefore its diagonals intersect each other, meaning:

AE=EC=12ACBE=ED=12BD AE=EC=\frac{1}{2}AC\\ BE=ED=\frac{1}{2}BD\\

b. A property of the rhombus is that its diagonals are perpendicular to each other, meaning:

AC⊥BD↕∢AEB=∢BEC=∢CED=∢DEA=90° AC\perp BD\\ \updownarrow\\ \sphericalangle AEB=\sphericalangle BEC=\sphericalangle CED=\sphericalangle DEA=90\degree

c. The definition of a rhombus - a quadrilateral where all sides are equal, meaning:

AB=BC=CD=DA AB=BC=CD=DA

Therefore, from the three facts mentioned in: a-c and using the SAS (Side-Angle-Side) congruence theorem, we can conclude that:

d.
△AEB≅△CEB≅△AED≅△CED \triangle AEB\cong\triangle CEB\cong\triangle AED\cong\triangle CED (where we made sure to properly and accurately match the triangles according to their vertices in correspondence with the appropriate sides and angles).

Indeed, we found that the diagonals of the rhombus create (together with the rhombus's sides - which are equal to each other) four congruent triangles.

Therefore - the correct answer is answer a.

Answer:

Yes

Frequently Asked Questions

Everything you need to know about Diagonals of a Rhombus

What are the 3 main properties of rhombus diagonals?

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The three key properties are: (1) diagonals intersect at their midpoints, (2) diagonals are perpendicular forming 90° angles, and (3) diagonals bisect the angles of the rhombus. These properties can be used directly without proof in problem-solving.

How do you find the area of a rhombus using diagonals?

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Use the formula: Area = (d₁ × d₂) ÷ 2, where d₁ and d₂ are the lengths of the diagonals. Multiply the diagonal lengths together, then divide by 2 to get the rhombus area.

Are the diagonals of a rhombus equal in length?

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No, the diagonals of a rhombus are not equal in length. Unlike a square, a rhombus has diagonals of different lengths, but they still intersect perpendicularly at their midpoints.

Why are rhombus diagonals always perpendicular?

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Rhombus diagonals are perpendicular because all four sides are equal, creating four congruent triangles when diagonals intersect. The corresponding angles must be right angles (90°) for the triangles to be congruent by SSS criterion.

How do rhombus diagonals bisect the vertex angles?

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Each diagonal divides the vertex angles into two equal parts. For example, if vertex A has angle measures, diagonal AC splits it so that ∠A₁ = ∠A₂. This occurs at all four vertices of the rhombus.

What congruent triangles are formed by rhombus diagonals?

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The intersecting diagonals create four congruent triangles. Each triangle shares equal sides (from the rhombus properties) and can be proven congruent using the Side-Side-Side (SSS) congruence criterion.

Can you solve for one diagonal if you know the area and other diagonal?

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Yes, rearrange the area formula: if Area = (d₁ × d₂) ÷ 2, then d₂ = (2 × Area) ÷ d₁. Simply multiply the area by 2, then divide by the known diagonal length.

What makes rhombus diagonal properties different from other quadrilaterals?

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Unlike general parallelograms, rhombus diagonals are always perpendicular and bisect vertex angles. Unlike rectangles, rhombus diagonals are unequal in length. These unique properties come from having four equal sides.

More Diagonals of a Rhombus Questions

Click on any question to see the complete solution with step-by-step explanations

Diagonals of a Rhombus

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