The vertices of the triangle shown in the graph are labeled as follows:

• $H(1, 8)$
• $I(7, -3)$
• $B(1, -10)$

To find the area of the triangle, we can apply the formula for the area of a triangle when the coordinates of the vertices are known:

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

Let the coordinates of the vertices be:

• $H(1, 8)$, so $x_1 = 1$, $y_1 = 8$
• $I(7, -3)$, so $x_2 = 7$, $y_2 = -3$
• $B(1, -10)$, so $x_3 = 1$, $y_3 = -10$

Substituting these values into the formula:

$\text{Area} = \frac{1}{2} \left| 1(-3 - (-10)) + 7(-10 - 8) + 1(8 - (-3)) \right|$ $= \frac{1}{2} \left| 1(7) + 7(-18) + 1(11) \right|$ $= \frac{1}{2} \left| 7 - 126 + 11 \right|$ $= \frac{1}{2} \left| -108 \right|$ $= \frac{1}{2} \times 108 = 54 \, \text{square units}$

Thus, the area of the triangle is 54 square units.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How can you verify the area of a triangle using another method, like determinants?
2. How does this formula work for any triangle with known vertices?
3. What happens to the area if one of the vertices is shifted horizontally or vertically?
4. Can you explain how the signs of the coordinates affect the calculation of the area?
5. How is the distance formula used to calculate side lengths in such triangles?

Tip: Always double-check your vertex coordinates before applying the area formula to avoid sign or position errors.