Question

How can we construct a rhombus inside a parallelogram in which the smaller side is two-third the larger side?

• A. Find the point which divides the larger side in the ratio of 2:3 and then draw a line parallel to smaller side.
• B. Measure with a scale and identify the point on the larger side and construct a rhombus.
• C. Using the length of the longer side construct a rhombus by extending the smaller side.
• D. Find the point which divides the larger side in the ratio of 2:1 and then draw a line parallel to smaller side.

Answer 1

The correct option is D Find the point which divides the larger side in the ratio of 2:1 and then draw a line parallel to smaller side.

A rhombus is a parallelogram with all sides equal.
To construct a rhombus inside a given parallelogram, we need to use the available lengths of the sides of the parallelogram.
The available sides are $l,\frac{2l}{3}$ (Considering ‘$l$’ to be the length of the longer side).
If we try to construct a rhombus with side $l$, it will lie outside the parallelogram.
So taking the side $\frac{2l}{3}$ to be one of the sides of the rhombus, we need to locate points on the longer side such that their length is $\frac{2l}{3}$.

Steps of Construction:
a) Let us start by constructing a ray AX which makes an acute angle with one of the longer sides AB.
b) Now mark $A_1,A_2,A_3$ on ray AX such that $A_1{A}_2=A_2{A}_3=A{A}_1$ . (This is done in order to divide $A{A}_2$ and $A_2{A}_3$ in the ratio 2:1.)

c) Now join $A_{3}B$ and draw $A_{2}E$ parallel to $A_{3}B$ such that the point E lies on the line segment AB. By Basic Proportionality Theorem, $\frac{AE}{EB}=\frac{A{A}_2}{A_2{A}_3}=\frac{2}{1}$.
d) Now draw a line from point E parallel to BC. The point where this line intersects CD be F.
e) AEFD is the required rhombus.