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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. 


Solution

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
4x + 2 = 18 cm
4x = 16 cm
x = 164 = 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.
DAB+ADC=180° (Co-interior angles are supplementary)
60°+ADC=180°
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. 

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