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प्रश्न
Construct a quadrilateral ABCD in which BC = 6 cm, AC = 7 cm, AD = 4.2 cm, ∠A = 120° and CD = 4 cm.
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उत्तर
Given:
BC = 6 cm
AC = 7 cm
AD = 4.2 cm
∠A = 120°
CD = 4 cm
Step-wise calculation (Construction steps):
1. Draw the diagonal AC of length 7 cm. ...(Draw segment AC = 7 cm)
2. Locate point D:
With centre A and radius 4.2 cm, draw an arc.
With centre C and radius 4 cm, draw an arc.
The intersection point(s) of these two arcs give the possible position(s) of D. Choose one intersection and label it D. ...(If the two arcs do not meet, there is no possible construction with the given data)
3. Construct the ray for AB, making ∠DAB = 120° at A:
At A, using AD as one side of the angle, construct the ray AX such that ∠DAX, i.e. ∠DAB = 120°.
Use compass-and-straightedge or a protractor to mark 120° from AD on the side where B is expected.
4. Locate point B:
With centre C and radius 6 cm, draw an arc circle with radius BC = 6 cm.
The intersection of this arc with the ray AX (From step 3) is point B.
If the ray meets the arc at two points, choose the one consistent with the chosen D there may be two mirror-image solutions.
5. Join A to B, B to C and C to D to complete the quadrilateral ABCD.
Notes
Step 2 ensures AD = 4.2 cm and CD = 4 cm simultaneously by using the two arcs intersection of circles.
Step 3 forces the required ∠A = 120°.
Step 4 forces BC = 6 cm.
These three constraints together give the required quadrilateral when intersections exist.
The same general method draw a diagonal, use arcs for side lengths and construct the required angle at a vertex is the standard approach for such constructions.
