Can we complete this quadrilateral so that all its sides are of the same length? Mark a point C whose distance from B and D is equal to (or ). To do this, measure using a compass. Keeping this length as the radius, cut arcs from B and D.

Understand the Question A quadrilateral having all four sides equal in length is a rhombus (). Given adjacent sides and with , we can locate the fourth vertex by drawing arcs of radius equal to from both and . The intersection of these two arcs marks point , completing the quadrilateral. Step 1 · Identify the Condition for Equal Sides For quadrilateral to have all sides of equal length, it must satisfy: A quadrilateral with four equal sides is a rhombus. Step 2 · Set the Compass Radius Measure the length of side by placing the compass needle at point and the pencil tip at point . Step 3 · Locate Point C Using Arcs Keeping the compass radius equal to : Place the compass needle at and draw an arc. Place the compass needle at and draw another arc intersecting the first arc. The point of intersection is point . Step 4 · Complete the Quadrilateral Join point to and point to using a ruler. Since , quadrilateral is a rhombus. Answer > Yes, we can complete this quadrilateral so that all its sides are of the same length; the resulting figure is a rhombus. Common Mistakes Varying the Compass Radius: Changing the compass opening between cutting arcs from and will lead to unequal side lengths (). Assuming It Must Be a Square: A quadrilateral with all sides equal is a rhombus in general; it is only a square if the angle at vertex is .

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Question 2

Can we complete this quadrilateral so that all its sides are of the same length?

Mark a point C whose distance from B and D is equal to $\text{AB}$ (or $\text{AD}$ ). To do this, measure $\text{AB}$ using a compass. Keeping this length as the radius, cut arcs from B and D.

Solution

Understand the Question

A quadrilateral having all four sides equal in length is a rhombus ( $\text{AB} = \text{BC} = \text{CD} = \text{DA}$ ).
Given adjacent sides $\text{AB}$ and $\text{AD}$ with $\text{AB} = \text{AD}$ , we can locate the fourth vertex $\text{C}$ by drawing arcs of radius equal to $\text{AB}$ from both $\text{B}$ and $\text{D}$ .
The intersection of these two arcs marks point $\text{C}$ , completing the quadrilateral.

Step 1 · Identify the Condition for Equal Sides

For quadrilateral $\text{ABCD}$ to have all sides of equal length, it must satisfy: $\text{AB} = \text{BC} = \text{CD} = \text{DA}$

A quadrilateral with four equal sides is a rhombus. Diagram 1

Step 2 · Set the Compass Radius

Measure the length of side $\text{AB}$ by placing the compass needle at point $\text{A}$ and the pencil tip at point $\text{B}$ . Diagram 2

Step 3 · Locate Point C Using Arcs

Keeping the compass radius equal to $\text{AB}$ :

Place the compass needle at $\text{B}$ and draw an arc.
Place the compass needle at $\text{D}$ and draw another arc intersecting the first arc.

The point of intersection is point $\text{C}$ . Diagram 3

Step 4 · Complete the Quadrilateral

Join point $\text{B}$ to $\text{C}$ and point $\text{D}$ to $\text{C}$ using a ruler. Diagram 4

Since $\text{AB} = \text{BC} = \text{CD} = \text{DA}$ , quadrilateral $\text{ABCD}$ is a rhombus.

Answer

Yes, we can complete this quadrilateral so that all its sides are of the same length; the resulting figure is a rhombus.

Common Mistakes

Varying the Compass Radius: Changing the compass opening between cutting arcs from $\text{B}$ and $\text{D}$ will lead to unequal side lengths ( $\text{BC} \neq \text{CD}$ ).
Assuming It Must Be a Square: A quadrilateral with all sides equal is a rhombus in general; it is only a square if the angle at vertex $\text{A}$ is $90^\circ$ .

Question 2

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