To solve this problem, let’s determine the coordinates of the vertices of the triangle (labeled $I$, $H$, and $G$) and use the formula for the area of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:

$\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$

Step 1: Identify coordinates

From the graph:

• $I = (-10, 12)$
• $H = (8, 4)$
• $G = (4, 2)$

Step 2: Apply the formula

Substitute the coordinates into the formula: $\text{Area} = \frac{1}{2} \left| (-10)(4 - 2) + 8(2 - 12) + 4(12 - 4) \right|$

Simplify each term: $= \frac{1}{2} \left| (-10)(2) + 8(-10) + 4(8) \right|$ $= \frac{1}{2} \left| -20 - 80 + 32 \right|$ $= \frac{1}{2} \left| -68 \right|$ $= \frac{1}{2} (68)$ $= 34$

Final Answer:

The area of the triangle is 34 square units.

Would you like more details or have any further questions?

Related Questions:

1. How is the formula for the area of a triangle derived?
2. Can this problem be solved geometrically without coordinates?
3. What are alternative methods for calculating the area of a triangle?
4. How does this formula generalize to 3D triangles?
5. Can you calculate the perimeter of the triangle using these coordinates?

Tip:

When working with geometric problems, always label and verify coordinates clearly from the graph to avoid calculation errors.