Given a rhombus $ABCD$ in which $AB = 4 \, cm$ and $\angle ABC = 60^{\circ}$,divide it into two triangles,$ABC$ and $ADC$. Construct a triangle $AB'C'$ similar to $\triangle ABC$ with a scale factor of $\frac{2}{3}$. Draw a line segment $C'D'$ parallel to $CD$,where $D'$ lies on $AD$. Is $AB'C'D'$ a rhombus? Give reasons.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) First,draw the rhombus $ABCD$ in which $AB = 4 \, cm$ and $\angle ABC = 60^{\circ}$ as shown in the figure,and join $AC$.
Construct the triangle $AB'C'$ similar to $\triangle ABC$ with a scale factor of $\frac{2}{3}$ as instructed in the Mathematics Textbook for Class $X$.
Finally,draw the line segment $C'D'$ parallel to $CD$ such that $D'$ lies on $AD$.
Since $\triangle AB'C' \sim \triangle ABC$,we have $\frac{AB'}{AB} = \frac{AC'}{AC} = \frac{B'C'}{BC} = \frac{2}{3}$.
Since $C'D' \parallel CD$,$\triangle AD'C' \sim \triangle ADC$. Thus,$\frac{AD'}{AD} = \frac{AC'}{AC} = \frac{C'D'}{CD} = \frac{2}{3}$.
Since $ABCD$ is a rhombus,$AB = BC = CD = AD$.
Therefore,$AB' = B'C' = C'D' = AD' = \frac{2}{3} AB$.
Since all four sides of the quadrilateral $AB'C'D'$ are equal,$AB'C'D'$ is a rhombus.

Explore More

Similar Questions

Write True or False and give reasons for your answer.
To construct a triangle similar to a given $\triangle ABC$ with its sides $\frac{7}{3}$ of the corresponding sides of $\triangle ABC$,draw a ray $BX$ making an acute angle with $BC$ and $X$ lies on the opposite side of $A$ with respect to $BC$. The points $B_{1}, B_{2}, \dots, B_{7}$ are located at equal distances on $BX$,$B_{3}$ is joined to $C$ and then a line segment $B_{6}C'$ is drawn parallel to $B_{3}C$ where $C'$ lies on $BC$ produced. Finally,line segment $A'C'$ is drawn parallel to $AC$.

Difficult
View Solution

Draw a triangle $ABC$ in which $AB = 4 \, cm$,$BC = 6 \, cm$,and $AC = 9 \, cm$. Construct a triangle similar to $\triangle ABC$ with a scale factor of $\frac{3}{2}$. Justify the construction. Are the two triangles congruent? Note that all three angles of the two triangles are equal,but the sides are not.

Difficult
View Solution

Draw a line segment of length $7 \, cm$. Find a point $P$ on it which divides it in the ratio $3: 5$.

$\odot(P, 5 \, cm)$ is given. Construct two tangents to the circle such that the measure of the angle between them is $120^\circ$. Write the steps of construction.

To divide a line segment $AB$ in the ratio $5:7,$ first a ray $AX$ is drawn so that $\angle BAX$ is an acute angle and then at equal distances points are marked on the ray $AX$ such that the minimum number of these points is

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo