Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of

(i) $30^\circ$

(ii) $40^\circ$

(iii) $90^\circ$

(iv) $140^\circ$

Question

Draw a quadrilateral whose diagonals have equal lengths of that bisect each other, and intersect at an angle of (i) (ii) (iii) (iv)

Homi · Accepted Answer

Understand the Question

A quadrilateral whose diagonals are equal and bisect each other is a rectangle.
If the diagonals are also perpendicular (intersect at $90^\circ$ ), the quadrilateral is a square.
To construct the quadrilateral:
1. Draw the first diagonal $AB = 8\text{ cm}$ and mark its midpoint $M$ such that $AM = MB = 4\text{ cm}$ .
2. At $M$ , draw a line segment making the given angle with $AB$ .
3. Mark points $C$ and $D$ along this line such that $MC = MD = 4\text{ cm}$ (making diagonal $CD = 8\text{ cm}$ ).
4. Join points $A, C, B, D$ to form the required quadrilateral $ACBD$ .

(i) Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of $30^\circ$ .

Step 1 · Draw First Diagonal and Find Midpoint

Draw a line segment $AB = 8\text{ cm}$ . Mark the midpoint $M$ on $AB$ such that: Diagram 1

\begin{aligned} AM = MB &= \dfrac{\text{Length of diagonal}}{2} \\[0.6em] &= \dfrac{8\text{ cm}}{2} = 4\text{ cm} \end{aligned}

Step 2 · Draw Second Diagonal at $30^\circ$

Using a protractor, draw a line through $M$ making an angle of $30^\circ$ with $MB$ . Diagram 3

On this line, mark points $C$ and $D$ on opposite sides of $M$ such that: Diagram 4

\begin{aligned} MC = MD &= \dfrac{8\text{ cm}}{2} = 4\text{ cm} \end{aligned}

Step 3 · Join Vertices to Complete Quadrilateral

Join $AC$ , $CB$ , $BD$ , and $DA$ to form the required quadrilateral $ACBD$ .Since diagonals $AB$ and $CD$ are equal ( $8\text{ cm}$ ) and bisect each other at $M$ , the resulting quadrilateral $ACBD$ is a rectangle.

Answer

(i) $ACBD$ is the required rectangle.

(ii) Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of $40^\circ$ .

Step 1 · Construct Diagonal $AB$ and Midpoint $M$

Draw line segment $AB = 8\text{ cm}$ . Mark midpoint $M$ such that $AM = MB = 4\text{ cm}$ .

Step 2 · Draw Diagonal $CD$ at $40^\circ$

Using a protractor, draw a line through $M$ making an angle of $40^\circ$ with $MB$ .

Mark points $C$ and $D$ on this line such that $MC = MD = 4\text{ cm}$ .

Step 3 · Join Vertices

Join $AC$ , $CB$ , $BD$ , and $DA$ to complete the quadrilateral $ACBD$ .

Since the diagonals are equal and bisect each other, $ACBD$ is a rectangle.

Answer

(ii) $ACBD$ is the required rectangle.

(iii) Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of $90^\circ$ .

Step 1 · Construct Diagonal $AB$ and Midpoint $M$

Draw line segment $AB = 8\text{ cm}$ . Mark midpoint $M$ such that $AM = MB = 4\text{ cm}$ .

Step 2 · Draw Perpendicular Diagonal $CD$

Using a protractor or compass, draw a perpendicular line through $M$ (at an angle of $90^\circ$ to $AB$ ).

Mark points $C$ and $D$ on this line such that $MC = MD = 4\text{ cm}$ .

Step 3 · Join Vertices

Join $AC$ , $CB$ , $BD$ , and $DA$ to obtain the quadrilateral $ACBD$ .

Since the diagonals are equal, bisect each other, and are perpendicular ( $90^\circ$ ), $ACBD$ is a square.

Answer

(iii) $ACBD$ is the required square.

(iv) Draw a quadrilateral whose diagonals have equal lengths of $8\text{ cm}$ that bisect each other, and intersect at an angle of $140^\circ$ .

Step 1 · Construct Diagonal $AB$ and Midpoint $M$

Draw line segment $AB = 8\text{ cm}$ . Mark midpoint $M$ such that $AM = MB = 4\text{ cm}$ .

Step 2 · Draw Diagonal $CD$ at $140^\circ$

Using a protractor, draw a line through $M$ making an angle of $140^\circ$ with $MB$ .

Mark points $C$ and $D$ on this line such that $MC = MD = 4\text{ cm}$ .

Step 3 · Join Vertices

Join $AC$ , $CB$ , $BD$ , and $DA$ to complete the quadrilateral $ACBD$ .

Since the diagonals are equal and bisect each other, $ACBD$ is a rectangle.

Answer

(iv) $ACBD$ is the required rectangle.

Common Mistakes

Marking Full Length from Midpoint: Measuring $8\text{ cm}$ on one side of $M$ instead of measuring $4\text{ cm}$ on both sides of $M$ .
Incorrect Quadrilateral Classification: Assuming that all quadrilaterals with equal bisecting diagonals are squares. Only when the angle of intersection is $90^\circ$ is the quadrilateral a square; otherwise, it is a rectangle.
Vertex Naming Order: Joining points in incorrect sequence (e.g., crossing lines $AB$ and $CD$ instead of joining perimeter vertices $A \to C \to B \to D \to A$ ).

Question 2

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