Question

Compute the area of the triangle using coordinates. What is the area of the triangle in square units?
Point A: (2, 4)
Point B: (9, 4)
Point C: (3, 2)

A. 3.5
B. 10
C. 7
D. 14

Answer 1

To compute the area of a triangle given its vertices in the coordinate plane, we can use the formula:

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

Where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the points A, B, and C.

Given:
- Point A: \( (2, 4) \)
- Point B: \( (9, 4) \)
- Point C: \( (3, 2) \)

Let:
- \( (x_1, y_1) = (2, 4) \)
- \( (x_2, y_2) = (9, 4) \)
- \( (x_3, y_3) = (3, 2) \)

Now, we substitute these coordinates into the area formula:

\[
\text{Area} = \frac{1}{2} \left| 2(4 - 2) + 9(2 - 4) + 3(4 - 4) \right|
\]
\[
= \frac{1}{2} \left| 2(2) + 9(-2) + 3(0) \right|
\]
\[
= \frac{1}{2} \left| 4 - 18 + 0 \right|
\]
\[
= \frac{1}{2} \left| -14 \right|
\]
\[
= \frac{1}{2} \times 14
\]
\[
= 7
\]

Thus, the area of the triangle is \( \boxed{7} \) square units.