Find the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1).

iohanetc

iohanetc

Answered question

2021-06-11

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1).

Answer & Explanation

un4t5o4v

un4t5o4v

Skilled2021-06-12Added 105 answers

I solved your question:

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-28Added 2605 answers

Consider the parallelogram A=(-3;0), B(-1,5), C(7;4) and D=(5;-1)

The objective is to find the area of parallelogram

Find the are as follows:

The area of the parallelogram is the magnitude of the cross product of the adjacent edges.

That is, Area=|AB×AD|

Find the adjacent edges AB and AD as follows

AB=BA

=(1;5)(3;0)

=(1+3;50)

=(2;5)

AD=DA

=(5;1)(3;0)

=(8;1)

Find AB×AD as,

AB×AD=[ijk250810]

=5(0)0(1)i(2(0)0(8))j+(2(1)5(8))k

=0i+0j42k

The area of the parallelogram is,

Area =|AB×AD|

=|(0i+0j42k|

=(0)2+(0)2+(42)2

=(42)2

= 42 square unit

Hence, the area of parallelogram is 42

Jeffrey Jordon

Jeffrey Jordon

Expert2023-04-30Added 2605 answers

The area of a parallelogram can be found by taking the cross product of its adjacent sides. Therefore, we can find the area of the parallelogram with vertices A(3,0), B(1,5), C(7,4), and D(5,1) as follows:
First, we need to find the two adjacent sides of the parallelogram. Let v be the vector from A to B and u be the vector from A to D. Then, we have:
v=[1(3)50]=[25]
u=[5(3)10]=[81]
Next, we take the cross product of v and u to find the area of the parallelogram:
v×u=|ijk250810|=[0042]
Therefore, the area of the parallelogram is |42|=42.
Vasquez

Vasquez

Expert2023-04-30Added 669 answers

To find the area of the parallelogram with vertices A(3,0), B(1,5), C(7,4), and D(5,1) is to use the distance formula to find the lengths of its adjacent sides.
Let ABCD be the parallelogram with vertices A(3,0), B(1,5), C(7,4), and D(5,1). Then, AB and AD are adjacent sides of the parallelogram.
Using the distance formula, we have:
AB=(xBxA)2+(yByA)2=(1(3))2+(50)2=20
AD=(xDxA)2+(yDyA)2=(5(3))2+(10)2=74
The area of the parallelogram can then be found by taking the product of the lengths of its adjacent sides and the sine of the angle between them:
Area=AB·AD·sin(θ)
where θ is the angle between vectors AB and AD. We can find θ using the dot product:
AB·AD=(xBxA)(xDxA)+(yByA)(yDyA)=(1(3))(5(1))+(50)(10)=42
Since AB·AD=AB·AD·cos(θ), we have:
cos(θ)=AB·ADAB·AD=422074=21185
Taking the sine of θ gives:
sin(θ)=1cos2(θ)=1(21185)2=8185
Therefore, the area of the parallelogram is:
Area=AB·AD·sin(θ)=20·74·8185=42

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