Answer:
23 square units
Step-by-step explanation:
You want the area of the triangle with vertices at grid points (0, 0), (-3, -4), and (-7, 6).
Area
There are several ways you can find the area of a triangle that is specified by grid points. A relatively simple one makes use of the "cross product" of two side vectors. We can write vectors for two of the sides as ...
AB = (-3, -4) -(0, 0) = (-3, -4)
AC = (-7, 6) -(0, 0) = (-7, 6)
The cross product of these is ...
AB × AC= (-3, -4) × (-7, 6) = (-3)(6) - (-4)(-7)
= -18 -28 = -46
The area is ...
Area = 1/2·|AB×AC|
Area = 1/2·|-46| = 46/2 = 23
The area of the triangle is 23 square units.
Pick's theorem
When the vertices are on grid points, the area can be found using the formula ...
A = i +b/2 -1
where i is the number of interior grid points, and b is the number of boundary grid points.
In the attachment, we have highlighted the interior grid points in purple, and the boundary grid points in black. Counting them, we have ...
A = 22 +4/2 -1 = 23
The area of the triangle is 23 square units.
Bounding rectangle
The triangle can be bounded by a rectangle 7 units wide and 10 units high. From that area, we can subtract the areas of the three triangles that are not included in the area of interest. Each of those has an area of ...
A = 1/2bh
So, the area of the triangle we want is ...
[tex]A = LW-\dfrac{1}{2}b_1h_1-\dfrac{1}{2}b_2h_2-\dfrac{1}{2}b_3h_3\\\\\\A=7\cdot10-\dfrac{1}{2}(7\cdot6+3\cdot4+4\cdot10)=70-\dfrac{42+12+40}{2}\\\\\\A=23[/tex]
The area of the triangle is 23 square units.
