How To Find The Fourth Vertex Of A Square
Surface area OF QUADRILATERAL WHEN Iv VERTICES ARE GIVEN
Permit us consider the quadrilateral ABCD shown below.
In the above quadrilateral, A(xone , y1 ), B( 102 , y2 ), C( x3 , yiii ) and D( xiv , y4 ) are the vertices.
To observe surface area of the quadrilateral ABCD, now we accept take the vertices A(x1 , yi ), B( 10ii , y2 ), C( x3 , y3 ) and D( xfour , y4 ) of the quadrilateral ABCD in order (counter clockwise management) and write them cavalcade-wise equally shown beneath.
Add the diagonal products x1ytwo , ten2 ythree , x3 y4 and ten4 yane are shown in the nighttime arrows.
(x1y2 + ten2y3+ x3y4 + 10fouryi) -----(one)
Add the diagonal products x2 yane , x3 y2 , x4 yiii and ten1 y4 are shown in the dotted arrows.
(x2y1+ x3y2+ x4ythree + xiyfour) -----(2)
Subtract (2) from (i) and multiply the deviation past ane/2 to get expanse of the quadrilateral ABCD.
So, area of the quadrilateral ABCD is
= (i/2) ⋅ { (x ane y 2 + x two y 3 + x 3 y 4 + ten 4 y one )
- (x 2 y 1 + x iii y ii + x 4 y three + x one y 4 ) }
Problem :
Find the area of the quadrilateral whose vertices are
(-4, -2), (-3, -5), (three, -ii) and (2, iii)
Solution :
Let A(-4, -2), B(-3, -v), C(3, -ii) and (2, 3).
Plot A, B, C and D in a crude diagram and take them in counter-clockwise order.
And then,
(x1 , y1 ) = (-4, -2)
(10ii, ytwo) = (-3, -5)
(ten3, y3) = (3, -ii)
(xiv, y4) = (two, 3)
Area of triangle ABC is
= (1/2) ⋅ { (x one y 2 + x 2 y three + 10 iii y 4 + 10 4 y 1 )
- (x 2 y 1 + 10 three y 2 + x 4 y iii + x 1 y 4 ) }
= (1/2) x {[20 + 6 + 9 - four] - [six - 15 - 4 - 12]}
= (i/2) ten {[31] - [-25] }
= (1/ii) ten {31 + 25}
= (1/two) x 56
= 28
So, expanse of the given quadrilateral is 28 square units.
Note :
If you get the expanse of a quadrilateral as a negative value, accept it as positive.
Because, the area of the quadrilateral is never negative. That is, we always accept the area of quadrilateral every bit positive.
Practice Questions
Discover the expanse of the each quadrilateral whose vertices are
(i) (6, nine), (seven, 4), (4, 2) and (iii, 7).
(ii) (-3, iv), (-five, -6), (4, -1) and (ane, 2)
(iii) (-iv, 5), (0, 7), (5, -5) and (-4, -2)
Answers :
(i) 17 foursquare units.
(ii) 43 square units
(iii) 60.5 foursquare units
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