Question

if one side of rhombus has end points (4,5) and (1,1), and maximum area of rhombus is
$k {}^{2}$
then k is

Answer 1

It is given that the co ordinates of end points of a side of rhombus are ( 4 , 5 ) and ( 1 , 1 ). And, the maximum area of the rhombus is k².


Now,
Let a rhombus ABCD, where D and C are the points with co ordinates ( 1 , 1 ) and ( 4 , 5 ) respectively.


Join the points A and C { AC is a diagonal }.

On observing the diagram, we get that the area of the rhombus is twice of the area of the ∆ADC,

• Then, draw a perpendicular E from A to DC and mark the angle between AD and DE as ∅.



Hence,
By Distance formula :

length of AE = √{ ( 5 - 1 )² + ( 4 - 1 )² }

length of AE = √( 4² + 3² )

length of AE = √25

length of AE = 5 { neglecting the -ve value of √25 }


And,
Length of AE should be hypotenuse ( i.e. side of rhombus, AD and it's length is 5, as discussed above ).

Therefore,
AE = 5 x sin∅



Hence,
Area of ∆ADC = 1 / 2 x ( DC ) x ( AE )

Area of ∆ADC = 1 / 2 x 5 units x ( 5 sin∅ )

Area of ∆ADC = 1 / 2 x 25 sin∅



From above,
Area of rhombus = 2 x area of ∆ACD

Area of rhombus = 2 x 1 / 2 x 25 sin∅

Area of rhombus = 25 sin∅



It is given that the maximum area of the rhombus is k². So the value of ∅ should 90°, as sin90° is equal to 1 i.e. the maximum value for any angle in terms of sine.


Thus,
K² = 25 x sin90°

K²= 25 x 1

K = √ 25

K = 5



Therefore the value of K is 5.
$\:$

Answer 2

Step-by-step explanation:

It is given that the co ordinates of end points of a side of rhombus are ( 4 , 5 ) and ( 1 , 1 ). And, the maximum area of the rhombus is k².