Question

37. The area of rhombus ABCD, whose vertices are taken in order are A=(-1, 1), B=(1,-2), C
=(3, 1) and D= (1, 4)
A) 6 sq. units
C) 18 sq. units
B) 12 sq. units
D) 24 sq. units
។​

Answer 1

Solution:

➛ Let A = ( -1, 1 )

➛ Let B = ( 1, -2 )

➛ Let C = ( 3, 1 )

➛ Let D = ( 1, 4 )

When we join A, B, C and D we will get a rhombus.

Steps:

• On a graph, draw the co-ordinate axes x0x' and y0y' intersecting at origin 0.

• With proper scale, mark the numbers on the two co-ordinate axis.

For plotting any point two steps are to be adopted:

• Starting from the origin 0 moves 1 unit along the negative side of the x-axis i.e, to the left of the origin 0.
• Now from here move 1 units that is parallel a positive direction of y-axis.
• Place a dot at the point reached.
• Label this as A ( -1, 1 ).

➤ Similarly plot the other points B ( 1, -2 ), C ( 3, 1 ) and D ( 1, 4 ).

Now, let's find out area:

✭ Area of rhombus = 1/2 ( product of its diagonals ✭

➥ Area of rhombus ABCD = 1/2 × AC × BD

We will find AC and BD using distance formula.

➥ AC = √((x₂ - x₁)² + (y₂ - y₁)²)

Here,

• x₁ = -1
• y₁ = 1
• x₂ = 3
• y₂ = 1

➥ AC = √(( 3 - (-1))² + (1 - 1)²)

➥ AC = √(( 3 + 1 )² + (1 - 1)²)

➥ AC = √(( 4 )² + (0)²)

➥ AC = √( 16 + 0)

➥ AC = √16

➥ AC = 4 units

Now, Finding BD

➥ BD = √((x₂ - x₁)² + (y₂ - y₁)²)

Here,

• x₁ = 1
• y₁ = -2
• x₂ = 1
• y₂ = 4

➥ BD = √(( 1 - 1)² + (4 - (-2))²)

➥ BD = √(( 0 )² + (4 + 2)²)

➥ BD = √(6)²

➥ BD = √36

➥ BD = 6 units

Now, put the values in the formula:

➥ Area of rhombus ABCD = 1/2 × AC × BD

➥ Area of rhombus ABCD = 1/2 × 4 × 6

➥ Area of rhombus ABCD = 1/2 × 24

➥ Area of rhombus ABCD = 24/2

➥ Area of rhombus ABCD = 12 sq. units

Option D) 24 sq. units is correct ✔

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