How to Construct a Rhombus When Given Side and Diagonals with Steps and Examples
Properties of a Rhombus
By definition, all the sides of a rhombus are congruent.
The angles are bisected with the help of diagonals.
The diagonals bisect each other perpendicularly. They are known as perpendicular bisectors.
All the properties of a parallelogram apply to the rhombus.
The opposite angles in the rhombus are always congruent to each other.
The consecutive angles in the rhombus are supplementary.
Rhombus Construction Steps
Let us say, we are asked to draw a rhombus LMNO with the length of its two diagonals. Let the diagonal LN = 4 cm and the diagonal OM = 5 cm. We must know the theorem that diagonals of a rhombus bisect each other at the right angle. Hence, we will not need any further dimensions to proceed with our construction of a rhombus.
Step 1:
Draw a vertical line segment LN of length 4 cm.
(image will be uploaded soon)
Step 2:
Place the compass at point L with any radius and from there draw an arc to the right and left respectively of the line segment LM. Similarly, repeat the process by placing the compass at point M. There should not be any change in the radius.
(image will be uploaded soon)
Step 3:
Mark the centre as P. Draw the second diagonal OM = 5 cm, to connect the two arc points intersecting each other so that PO = PM = 7 / 2 = 3.5 cm
(image will be uploaded soon)
Step 4:
Now join the points L with M, M with N, N with O, and O with L.
(image will be uploaded soon)
Then, LMNO is the required rhombus with the diagonals measuring 4 cm and 5 cm respectively. Let us keep in mind that P is the center of the two diagonals and < LPM = < MPN \[\leq\] NPO \[\leq\] OPL = 90 degrees.
FAQs on Construction of a Rhombus Using Compass and Ruler
1. What is a rhombus in geometry?
A rhombus is a quadrilateral in which all four sides are equal in length. It is a type of parallelogram, so its opposite sides are parallel and opposite angles are equal. Key properties include:
- All sides are equal.
- Opposite angles are equal.
- Diagonals bisect each other at right angles.
- Diagonals bisect the interior angles.
2. How do you construct a rhombus when one side and one angle are given?
To construct a rhombus when one side and one angle are given, draw all sides equal and maintain the given angle at one vertex. Follow these steps:
- Draw line segment AB equal to the given side.
- At point A, construct the given angle (for example, 60°).
- On the angle ray, mark point D such that AD = AB.
- With centers B and D and radius equal to AB, draw arcs intersecting at point C.
- Join BC and CD.
3. How do you construct a rhombus when both diagonals are given?
To construct a rhombus when both diagonals are given, use the property that diagonals bisect each other at right angles. Steps:
- Draw diagonal AC equal to the given length.
- Find its midpoint O.
- At O, draw a perpendicular line.
- On this perpendicular, mark points B and D such that OB = OD = half of the second diagonal.
- Join A, B, C, and D in order.
4. What tools are required for the construction of a rhombus?
The construction of a rhombus requires basic geometric tools: ruler, compass, and protractor. These tools help to:
- Draw equal line segments using a compass.
- Measure and construct given angles using a protractor.
- Draw straight lines with a ruler.
5. What are the properties used in constructing a rhombus?
The construction of a rhombus is based on its key geometric properties. These include:
- All sides are equal.
- Opposite sides are parallel.
- Diagonals bisect each other at 90°.
- Diagonals bisect the interior angles.
6. What is the formula for the area of a rhombus?
The area of a rhombus is given by the formula Area = (d₁ × d₂) / 2, where d₁ and d₂ are the diagonals. For example:
- If d₁ = 10 cm and d₂ = 8 cm,
- Area = (10 × 8) / 2 = 80 / 2 = 40 cm².
7. How do you prove that a constructed figure is a rhombus?
A constructed figure is a rhombus if all four sides are equal or if its diagonals bisect each other at right angles. To verify:
- Measure all four sides to confirm equality.
- Check that opposite sides are parallel.
- Verify that diagonals intersect at 90° and bisect each other.
8. What is the difference between a rhombus and a square?
The main difference is that a square has all angles equal to 90°, while a rhombus does not necessarily have right angles. Comparison:
- Both have all sides equal.
- A square has four right angles.
- A rhombus has equal sides but angles may be acute or obtuse.
9. Can a rhombus be constructed if only one side is given?
No, a rhombus cannot be uniquely constructed with only one side given because an angle or diagonal is also required. Since many rhombuses can have the same side length but different angles, additional information such as:
- One interior angle, or
- Lengths of diagonals
10. What are common mistakes in the construction of a rhombus?
Common mistakes in constructing a rhombus include not ensuring that all four sides are equal and ignoring perpendicular diagonals. Frequent errors:
- Incorrect compass width while marking equal sides.
- Failure to draw diagonals at 90° when required.
- Incorrect measurement of the given angle.
