PROBLEMS ON PROPERTIES OF RHOMBUS

  • All sides of rhombus are equal.
  • The opposite sides of a rhombus are parallel.
  • Opposite angles of a rhombus are equal.
  • In a rhombus, diagonals bisect each other at right angles.
  • Diagonals bisect the angles of a rhombus.
  • The sum of two adjacent angles is equal to 180 degrees.
  • The two diagonals of a rhombus form four right angled triangles which are congruent to each other.

Problem 1 :

Three vertices of a rhombus taken in order are 

(2, −1), (3, 4) and (−2, 3) 

find the fourth vertex.

Solution :

Let the given points be A (2, −1), B (3, 4) and C (−2, 3) and the required point be D (a, b).

Midpoint of diagonal AC  =  Midpoint of diagonal BD

Midpoint of diagonal AC  =  (x+ x2)/2, (y+ y2)/2

Midpoint of diagonal AC  =  (2 - 2)/2, (-1 + 3)/2

  =  (0, 1) ----(1)

Midpoint of diagonal BD  =  (3 + a)/2, (4 + b)/2 ----(2)

(1)  =  (2)

(3 + a)/2, (4 + b)/2  =  (0, 1)

Equating x coordinate

(3 + a)/2  =  0

3 + a  =  0

a  =  -3

Equating y coordinate

 (4 + b)/2  =  1

4 + b  =  2

b  =  -2

So, the required vertex is (-3, -2).

Problem 2 :

Assume quadrilateral EFGH is a rhombus. If the perimeter of EFGH is 24 and the length of diagonal EG = 10, what is the length of diagonal FH?

Solution :

Perimeter of rhombus  =  24

4a  =  24

a  =  6

Side length of rhombus is 6 

Length of diagonal  =  10

In rhombus, we have four right triangles.

In triangle OEF

OE  =  5

Half length of diagonal FH  =  x

102  =  52 + x2

100 - 25  =  x2

x=  75

x  =  5√3

Problem 3 :

If the area of a rhombus is 112 cm2 and one of its diagonal is 14 cm find its other.

Solution :

Area of rhombus  =  (d1 ⋅ d2)/2

d1  =  14 cm, d =  ?

Area  =  112 cm2

 (14 ⋅ d2)/2  =  112

d2  =  112(2)/14

d2  =  16

So, the other diagonal is 16 cm.

Problem 4 :

The length of diagonal are ratio 5:4 area of rhombus is 2250 cm2 find the length of diagonals.

Solution :

Length of diagonals are 5x and 4x.

Area of rhombus  =  2250 cm2

 (5x ⋅ 4x)/2  =  2250

20x =  4500

x =  225

x  =  15 cm

5x  =  5(15)  ==>  75

4x  =  4(15)  ==>  60

Length of diagonals are 75 cm and 60 cm.

Problem 5 :

If opposite angles of a rhombus are (2x)° and (3x-40)° then value of x is ?

Solution :

In a rhombus, opposite angles will be equal. 

2x  =  3x-40

x  =  40

Problem 6 :

ABCD is a rhombus in which AB is 3x-2, AC is 4x+4 and BD is 2x. Find x.

Solution :

AB  =  AD  =  3x-2

AC/2  =  (4x + 4)/2 ==>  2x + 2

BD/2  =  2x/2 ==>  2

(3x - 2)2  =  (2x + 2)2 + x2

9x+ 4 - 12x  =  4x+ 8x + 4 + x2

4x- 12x - 8x + 4 - 4  =  0

4x- 20x  =  0

4x(x - 5)  =  0

x can not be 0.

So, x  =  5.

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