page rhombus is a=7,the opposite sides are A(4,9) and B(-2,1).Calculate the area of a rhombus

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should notice that the problem provides the coordinates of two opposite vertices `A` `(4;9)` and `B(-2;1)` , hence, you may evaluate the length of diagonal of rhombus, thus, you need to use the diagonal formula to evaluate the area, such that:

`A = (d_1*d_2)/2`

You may evaluate `d_1 = AB` , using the distance formula, such that:

`AB = sqrt((x_B - x_A)^2 + (y_B - y_A)^2)`

`AB = sqrt((-2 - 4)^2 + (1 - 9)^2)`

`AB = sqrt(36 + 64) => AB = sqrt100 => AB = 10`

You should notice that the problem provides the lengths of side of rhombus, a = 7, hence, using the information that the diagonals of rhombus are perpendicular, you may find one half of diagonal `d_2` , such that:

`(d_2)/2 = sqrt(a^2 - ((d_1)/2)^2)`

`(d_2)/2= sqrt (49 - 25) => (d_2)/2 = sqrt 24 => (d_2)/2 = 2sqrt 6`

`d_2 = 4sqrt 6`

You may evaluate now the area of the rhombus, such that:

`A = (10*4sqrt 6)/2 => A = 20sqrt 6`

Hence, evaluating the area of the rhombus, using the diagonal formula, yields `A = 20sqrt 6.`

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