Q23 of 184 Page 10

Show that A(-3,2), B(-5,-5), C(2,-3) and D(4,4) are the vertices of a rhombus.

Note that to show that a quadrilateral is a rhombus, it is sufficient to show that


(a) ABCD is a parallelogram, i.e., AC and BD have the same midpoint.


(b) a pair of adjacent edges are equal


(c) the diagonal AC and BD are not equal.



Let A(-3, 2), B(-5,-5), C(2,-3) and D(4,4) are the vertices of a rhombus.


Coordinates of the midpoint of AC are



Coordinates of the midpoint of BD are



Thus, AC and BD have the same midpoint.


Hence, ABCD is a parallelogram


Now, using Distance Formula


d(A,B)= AB = √(-5 + 3)2 + (-5 – 2)2


⇒ AB = √(-2)2 + (-7)2


⇒ AB = √4 +49


⇒ AB = √53 units


d(B,C)= BC = √(-5 – 2)2 + (-5 + 3)2


⇒ BC = √(-7)2 + (-2)2


⇒ BC = √49 +4


⇒ BC = √53 units


d(C,D) = CD = √(4 – 2)2 + (4 + 3)2


⇒ CD = √(2)2 + (7)2


⇒ CD = √4 +49


⇒ CD = √53 units


d(A,D) = AD =√(4 + 3)2 +(4 – 2)2


⇒ AD = √(7)2 + (2)2


⇒ AD = √49 +4


⇒ AD = √53 units


Therefore, AB = BC = CD = AD = √53 units


Now, check for the diagonals


AC = √(2 + 3)2 + (-3 – 2)2


= √(5)2 + (-5)2


= √25 + 25


= √50


and


BD = √(4 + 5)2 + (4 + 5)2


⇒ BD = √(9)2 + (9)2


⇒ BD = √81 + 81


⇒ BD = √162


⇒ Diagonal AC ≠ Diagonal BD


Hence, ABCD is a rhombus.


More from this chapter

All 184 →
21

Find the centroid of the triangle ABC whose vertices are A (9,2), B(1,10) and C(-7,-6). Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

 22

If (1,2), (0,-1) and (2,-1) are the middle points of the sides of the triangle, find the coordinates of its centroid.

 24

Show that the point (3,2),(0,5),(-3,2) and (0,-1) are the vertices of a square.

 25

Prove that the points (-2,-1), (1,0),(4,3) and (1,2) are the vertices of a parallelogram.