Question

what is the area of δefg?
coordinate grid with y-axis (range -10 to 10) and x-axis (range -10 to 10), points e, f, g plotted. below: area = \square square units

Explanation:

Step1: Determine coordinates of points

From the graph, we can identify the coordinates of the vertices of \(\triangle EFG\). Let's assume the grid has each square with side length 1 unit.
- Point \(E\): Let's say its coordinates are \((3, 7)\) (by counting the grid lines: 3 units right on x - axis, 7 units up on y - axis)
- Point \(G\): Coordinates \((3, - 4)\) (3 units right on x - axis, 4 units down on y - axis)
- Point \(F\): Coordinates \((5, - 2)\) (5 units right on x - axis, 2 units down on y - axis)

Step2: Calculate the base length

The base of the triangle can be the vertical distance between \(E\) and \(G\) since they have the same \(x\) - coordinate (\(x = 3\)). The length of \(EG\) is calculated by the difference in \(y\) - coordinates.
The formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) when \(x_1=x_2\) is \(d=\vert y_1 - y_2\vert\).
For \(E(3,7)\) and \(G(3, - 4)\), \(EG=\vert7-(-4)\vert=\vert7 + 4\vert=11\)? Wait, no, maybe we can use another approach. Alternatively, we can use the formula for the area of a triangle using coordinates. The formula for the area of a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) is \(A=\frac{1}{2}\vert x_1(y_2 - y_3)+x_2(y_3 - y_1)+x_3(y_1 - y_2)\vert\)

Let's assign:
\(x_1 = 3,y_1 = 7\) (Point \(E\))
\(x_2 = 3,y_2=-4\) (Point \(G\))
\(x_3 = 5,y_3=-2\) (Point \(F\))

Substitute into the formula:
\(A=\frac{1}{2}\vert3(-4-(-2))+3(-2 - 7)+5(7-(-4))\vert\)
\(=\frac{1}{2}\vert3(-4 + 2)+3(-9)+5(11)\vert\)
\(=\frac{1}{2}\vert3(-2)-27 + 55\vert\)
\(=\frac{1}{2}\vert-6-27 + 55\vert\)
\(=\frac{1}{2}\vert22\vert\)
\( = 11\)? Wait, that seems incorrect. Maybe we made a mistake in coordinates. Let's re - check the coordinates.

Looking at the graph again:
- Point \(E\): Let's count the grid. From the origin (0,0), moving 3 units right (x = 3) and 7 units up (y = 7), so \(E(3,7)\)
- Point \(G\): Moving 3 units right (x = 3) and 4 units down (y=-4), so \(G(3, - 4)\)
- Point \(F\): Moving 5 units right (x = 5) and 2 units down (y=-2), so \(F(5, - 2)\)

Another way: The base can be the horizontal distance between \(G\) and \(F\) or the vertical distance. Wait, maybe we can use the method of enclosing the triangle in a rectangle.

The rectangle that encloses \(\triangle EFG\) would have:
- Left \(x\) - coordinate: 3 (from \(E\) and \(G\))
- Right \(x\) - coordinate: 5 (from \(F\))
- Bottom \(y\) - coordinate: - 4 (from \(G\))
- Top \(y\) - coordinate: 7 (from \(E\))

The length of the rectangle (horizontal side) is \(5 - 3=2\)
The height of the rectangle (vertical side) is \(7-(-4)=11\)
The area of the rectangle is \(2\times11 = 22\)

Now, the triangle \(\triangle EFG\) is half of the rectangle? Wait, no. Wait, if we look at the coordinates, the base can be the length between \(G\) and \(E\) (vertical line) with length \(7-(-4)=11\), and the horizontal distance from this vertical line to \(F\) is \(5 - 3 = 2\). The area of a triangle is \(\frac{1}{2}\times base\times height\), where base is the vertical length (\(11\)) and height is the horizontal distance (\(2\))? Wait, no, that's not the right way. Wait, actually, the base should be the length of the side, and the height is the perpendicular distance from the opposite vertex to that side.

Let's take \(EG\) as the base. The length of \(EG\): since \(E(3,7)\) and \(G(3, - 4)\), the length \(EG=\vert7-(-4)\vert = 11\). The slope of \(EG\) is undefined (vertical line), so the perpendicular distance from \(F(5, - 2)\) to the line \(x = 3\) is the horizontal distance between \(x = 5\) and \(x = 3\), which is \(5 - 3=2\). Then the area of \(\triangle EFG\) is \(\frac{1}{2}\times base\times height=\frac{1}{2}\times11\times2 = 11\)? Wait, but let's check with another method.

Using the shoelace formula:
List the coordinates in order: \(E(3,7)\), \(G(3, - 4)\), \(F(5, - 2)\), and back to \(E(3,7)\)

Shoelace formula: \(A=\frac{1}{2}\vert\sum_{i = 1}^{n}(x_iy_{i + 1}-x_{i+1}y_i)\vert\)

For \(i = 1\): \(x_1 = 3,y_1 = 7\); \(x_2 = 3,y_2=-4\). So \(x_1y_2 - x_2y_1=3\times(-4)-3\times7=-12 - 21=-33\)

For \(i = 2\): \(x_2 = 3,y_2=-4\); \(x_3 = 5,y_3=-2\). So \(x_2y_3 - x_3y_2=3\times(-2)-5\times(-4)=-6 + 20 = 14\)

For \(i = 3\): \(x_3 = 5,y_3=-2\); \(x_4 = 3,y_4=7\). So \(x_3y_4 - x_4y_3=5\times7-3\times(-2)=35 + 6 = 41\)

Sum these values: \(-33+14 + 41=22\)

Then the area \(A=\frac{1}{2}\vert22\vert = 11\)

Wait, but maybe we made a mistake in coordinates. Let's re - examine the graph. Let's count the grid squares again.

Looking at the graph:
- Point \(E\): Let's see, the x - coordinate: from 0, moving 3 units to the right (x = 3), y - coordinate: moving 7 units up (y = 7). Correct.
- Point \(G\): x = 3, y=-4 (4 units down from 0). Correct.
- Point \(F\): x = 5, y=-2 (2 units down from 0). Correct.

Alternatively, we can use the formula for the area of a triangle with base and height. The vertical distance between \(E\) and \(G\) is \(7-(-4)=11\) (this is the length of the vertical segment \(EG\)). The horizontal distance from \(F\) to the line \(EG\) (which is \(x = 3\)) is \(5 - 3 = 2\). Since the line \(EG\) is vertical, the horizontal distance is the height corresponding to the base \(EG\). Then the area is \(\frac{1}{2}\times base\times height=\frac{1}{2}\times11\times2 = 11\).

Wait, but let's check with a different approach. Let's find the lengths of the sides and use Heron's formula.

First, find the lengths of \(EG\), \(GF\), and \(EF\).

Length of \(EG\): between \(E(3,7)\) and \(G(3, - 4)\): \(d_{EG}=\sqrt{(3 - 3)^2+(7-(-4))^2}=\sqrt{0 + 121}=11\)

Length of \(GF\): between \(G(3, - 4)\) and \(F(5, - 2)\): \(d_{GF}=\sqrt{(5 - 3)^2+(-2-(-4))^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\)

Length of \(EF\): between \(E(3,7)\) and \(F(5, - 2)\): \(d_{EF}=\sqrt{(5 - 3)^2+(-2 - 7)^2}=\sqrt{4 + 81}=\sqrt{85}\)

Now, using Heron's formula: \(s=\frac{a + b + c}{2}=\frac{11+2\sqrt{2}+\sqrt{85}}{2}\). This seems complicated, and our previous result of 11 is more straightforward, so we can trust the shoelace formula and the base - height method.

Answer:

11