To find the area of a triangle given the coordinates of its vertices, we use the coordinate geometry formula:

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

Step 1: Assigning the coordinates

• $A(30, 22.5)$
• $B(17.5, 35.625)$
• $C(17.5, 13.125)$

Let:

• $(x_1, y_1) = (30, 22.5)$
• $(x_2, y_2) = (17.5, 35.625)$
• $(x_3, y_3) = (17.5, 13.125)$

Step 2: Apply the formula

$\text{Area} = \frac{1}{2} \left| 30(35.625 - 13.125) + 17.5(13.125 - 22.5) + 17.5(22.5 - 35.625) \right|$

First, calculate the individual terms inside the absolute value:

$35.625 - 13.125 = 22.5$ $13.125 - 22.5 = -9.375$ $22.5 - 35.625 = -13.125$

Now substitute these results into the formula:

$\text{Area} = \frac{1}{2} \left| 30(22.5) + 17.5(-9.375) + 17.5(-13.125) \right|$

$30(22.5) = 675, \quad 17.5(-9.375) = -164.0625, \quad 17.5(-13.125) = -229.6875$

$\text{Area} = \frac{1}{2} \left| 675 - 164.0625 - 229.6875 \right|$

$\text{Area} = \frac{1}{2} \left| 675 - 393.75 \right| = \frac{1}{2} \left| 281.25 \right|$

$\text{Area} = \frac{1}{2} \times 281.25 = 140.625$

Final Answer

The area of the triangle is:

$\boxed{140.625} \, \text{square units}$

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Do you have any questions or need further clarifications?

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Here are some related questions:

1. How is the coordinate formula for a triangle’s area derived?
2. Can the area of a triangle be negative using coordinates?
3. How do you calculate the centroid of a triangle with given vertices?
4. What if one or more points are collinear—what will the area be?
5. How does this method relate to determinants of matrices?

Tip: If the three points are collinear, the area will always be zero since they form a straight line.