Question

The total perimeter of a quadrilateral ABCD is 18 cm. The opposite sides are equal and parallel. The adjacent sides are 1 cm more than the other. One of the angles of the quadrilateral ABCD measures 60°. Construct the quadrilateral ABCD.

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Given,

• Perimeter of ABCD = 18 cm... (i)
• Opposite sides are equal and parallel...(ii)
• Adjacent sides are 1 cm more than the other...(iii)
• One angle measure 60°...(iv)

Let us consider a rough sketch of ABCD.
AB and BC are adjacent sides and measure of AB = x, the measure of CD = x + 1.

From (i), (ii) and (iii),
AB + BC + CD + DA = 18cm
x + x + 1 + x + x + 1 = 18 cm
$\Rightarrow$ 4x + 2 = 18 cm
$\Rightarrow$ 4x = 16 cm
$\Rightarrow$ x = $\frac{16}{4}$ = 4 cm.

Then, AB = 4 cm, CD = 4 + 1 = 5 cm.
Now, CD = 4 cm and BC = 5 cm.

From (ii), ABCD is a parallelogram.
Hence, AB || CD.
$\angle DAB+\angle ADC=180°$ (Co-interior angles are supplementary)
$\Rightarrow 60°+\angle ADC=180°$
$\Rightarrow \angle ADC=120°$

Steps of construction:

Step 1: Draw a line segment AB of length 4 cm
Step 2: Construct an angle of 60° at the point A and extend the line.
Step 3: Keeping A as the centre, draw an arc of radius 5 cm on the extended line and mark it as D.
Step 4: Construct 120° at D and extend the line.
Step 5: Keeping D as the centre, draw an arc of radius 4 cm on the extended line and mark it as C.
Step 6: Join B and C.

After construction, we get the following quadrilateral.