find the area of a triangle ABC if coordinates of A( 30,22.5),B(17.5,35.625) and C (17.5,13.125)

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.

Find the area of a triangle ABC if coordinates of A(30, 22.5), B(17.5, 35.625), and C(17.5, 13.125).

Learn how to find the area of a triangle using coordinate geometry. This step-by-step solution involves applying the coordinate formula for the area, with vertices at A(30, 22.5), B(17.5, 35.625), and C(17.5, 13.125). Suitable for students in Grades 9-11.

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