The points $A (2, 0), B (9, 1), C (11, 6)$ and $D (4, 4)$ are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.
Given:
The points $A (2, 0), B (9, 1), C (11, 6)$ and $D (4, 4)$ are the vertices of a quadrilateral ABCD.
To do:
We have to determine whether ABCD is a rhombus or not.
Solution:
We know that,
The distance between two points \( \mathrm{A}\left(x_{1}, y_{1}\right) \) and \( \mathrm{B}\left(x_{2}, y_{2}\right) \) is \( \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \).
Therefore,
\( A B=\sqrt{(9-2)^{2}+(1-0)^{2}} \)
\( =\sqrt{(7)^{2}+(1)^{2}} \)
\( =\sqrt{49+1} \)
\( =\sqrt{50} \)
\( B C=\sqrt{(11-9)^{2}+(6-1)^{2}} \)
\( =\sqrt{(2)^{2}+(5)^{2}} \)
\( =\sqrt{4+25} \)
\( =\sqrt{29} \)
\( C D=\sqrt{(4-11)^{2}+(4-6)^{2}} \)
\( =\sqrt{49+4} \)
\( =\sqrt{53} \)
\( D A=\sqrt{(4-2)^{2}+(4-0)^{2}} \)
\( =\sqrt{(2)^{2}+(4)^{2}} \)
\( =\sqrt{4+16} \)
\( =\sqrt{20} \)
Here,
\( \mathrm{AB}≠\mathrm{BC}≠\mathrm{CD}≠\mathrm{DA} \)
Therefore, the given quadrilateral is not a rhombus.
Related Articles
- Find the area of quadrilateral ABCD, whose vertices are: $A( -3,\ -1) ,\ B( -2,\ -4) ,\ C( 4,\ -1)$ and$\ D( 3,\ 4) .$
- Find the area of a quadrilateral $ABCD$, the coordinates of whose vertices are $A (-3, 2), B (5, 4), C (7, -6)$ and $D (-5, -4)$.
- If the points $A (a, -11), B (5, b), C (2, 15)$ and $D (1, 1)$ are the vertices of a parallelogram $ABCD$, find the values of $a$ and $b$.
- Show that the points $A( 2,\ -1),\ B( 3,\ 4),\ C( -2,\ 3)$ and $D( -3,\ -2)$ are the vertices of a rhombus.
- If $A (-3, 5), B (-2, -7), C (1, -8)$ and $D (6, 3)$ are the vertices of a quadrilateral $ABCD$, find its area.
- Prove that the points $A (2, 3), B (-2, 2), C (-1, -2)$ and $D (3, -1)$ are the vertices of a square $ABCD$.
- Prove that the points $A (1, 7), B (4, 2), C (-1, -1)$ and $D (-4, 4)$ are the vertices of a square.
- If $A( –2,\ 1), B( a,\ 0), C( 4,\ b) $and $D( 1,\ 2)$ are the vertices of a parallelogram ABCD, find the values of a and b, Hence find the lengths of its sides.
- $ABCD$ is a rectangle formed by the points $A(-1, -1), B(-1, 4), C(5, 4)$ and $D(5, -1), P, Q, R$ and $S$ are the mid-points of $AB, BC, CD$ and $DA$ respectively. Is the quadrilateral $PQRS$ a square? a rectangle? or a rhombus? Justify your answer.
- Show that the points $(-4, -1), (-2, -4), (4, 0)$ and $(2, 3)$ are the vertices points of a rectangle.
- In a quadrilateral $ABCD$, the angles $A, B, C$ and $D$ are in the ratio $1 : 2 : 4 : 5$. Find the measures of each angle of the quadrilateral.
- $ABCD$ is a rectangle formed by joining the points $A (-1, -1), B (-1, 4), C (5, 4)$ and $D (5, -1). P, Q, R$ and $S$ are the mid-points of sides $AB, BC, CD$ and $DA$ respectively. Is the quadrilateral $PQRS$ a square? a rectangle? or a rhombus? Justify your answer.
- Prove that the points $(-4, -1), (-2, -4), (4, 0)$ and $(2, 3)$ are the vertices of a rectangle.
- Show that the points $A (1, 0), B (5, 3), C (2, 7)$ and $D (-2, 4)$ are the vertices of a parallelogram.
- Find the area of a parallelogram $ABCD$ if three of its vertices are $A (2, 4), B (2 + \sqrt3, 5)$ and $C (2, 6)$.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies