find the area of the pentagon whose vertices have co -ordinate (4,3),(-5,6),(0,-7),(3,-6)&(-7,-2).
To find the area of a pentagon given its vertices, you can use the Shoelace formula. The Shoelace formula calculates the signed area of a polygon using the coordinates of its vertices.
Here's how you can use the Shoelace formula to find the area of the pentagon with the given vertices:
Step 1: Arrange the vertices in the counterclockwise order
The given vertices are:
A = (4,3)
B = (-5,6)
C = (0,-7)
D = (3,-6)
E = (-7,-2)
To calculate the signed area, you need to arrange the vertices in a counterclockwise order. Start with any vertex and go around the pentagon in a counterclockwise direction. Let's arrange the vertices as follows:
A → B → C → D → E → A
Step 2: Apply the Shoelace formula
Using the Shoelace formula, the signed area (A) can be calculated as:
A = (1/2) * |(x1 * y2 + x2 * y3 + ... + xn * y1) - (y1 * x2 + y2 * x3 + ... + yn * x1)|
In this case, the formula becomes:
A = (1/2) * |(4 * 6 + (-5) * (-7) + 0 * (-6) + 3 * (-2) + (-7) * 3) - (3 * (-5) + (-7) * 0 + (-6) * 3 + (-2) * (-7) + 4 * 3)|
Simplifying the formula, we get:
A = (1/2) * |(24 + 35 + 0 - 6 - 21) - (-15 + 0 - 18 + 14 + 12)|
A = (1/2) * |(32 + 15) - (-7)|
A = (1/2) * |47 + 7|
A = (1/2) * 54
A = 27
Therefore, the area of the pentagon is 27 square units.
I would re-list them in counterclockwise order:
(4,3),(-5,6),(-7,-2),(0,-7),(3,-6)
Now list them in a column, in order, then repeating the first one listed
(4,3)
(-5,6)
(-7,-2)
(0,-7)
(3,-6)
(4,3)
area = (1/2)| sum of downproducts - sum of upproduct|
= (1/2)|(24+10 + 49+0+9) - (-15-42+0-21-24)|
= 97
check my arithmetic
see
http://www.mathopenref.com/coordpolygonarea.html