Question

Question 5
Construct the following and give justification :​​​​​​​
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A rhombus whose diagonals are 4cm and 6cm in lengths.

Answer 1

We know that all sides of a rhombus are equal and the diagonals of a rhombus are perpendicular bisectors of one another.
So, to construct a rhombus whose diagonals are 4cm and 6cm use the following steps.
(i) Draw the diagonal say AC = 4cm
(ii) Taking A and C as centres and radius more than $\frac{1}{2}AC$draw arcs on both sides of the line segment AC to intersect each other.
(iii) Let both the arcs intersect each other at P and Q, then join PQ.
(iv) Let PQ intersect AC at the point O. Thus, PQ is the perpendicular bisector of AC.
(v) Cut off 3cm lengths on OP and OQ, and mark those points as B and D.
(vi) Now, join AB, BC, CD and DA.
Thus, ABCD is the required rhombus.

Justification
Since D and B lie on the perpendicular bisector of AC. DA = DC and BA = BC ……(i)
[since every point on perpendicular bisector of line segment is equidistant from end points of line segment]
Now, $\angle DOC={90}^{\circ }$ Also, OD = OB = 3cm
Thus, AC is perpendicular bisector or BD.

CD = CB

From Eqs.(i) and (ii), AB = BC = CD = DA ….(ii)
Hence, ABCD is a rhombus.