Question

What is the area of the triangle in square units?

(1 point)
Responses

15.5
15.5

20
20

40
40

10
A=(3,1)
B=(3,9)
C=(8,8)

Answer 1

To find the area of triangle ABC with vertices A(3, 1), B(3, 9), and C(8, 8), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

Plugging in the coordinates:

- \(A(3, 1)\) → \(x_1 = 3, y_1 = 1\)
- \(B(3, 9)\) → \(x_2 = 3, y_2 = 9\)
- \(C(8, 8)\) → \(x_3 = 8, y_3 = 8\)

Now, substituting these values into the formula:

\[
\text{Area} = \frac{1}{2} \left| 3(9 - 8) + 3(8 - 1) + 8(1 - 9) \right|
\]

Calculating each term:

1. \(3(9 - 8) = 3 \cdot 1 = 3\)
2. \(3(8 - 1) = 3 \cdot 7 = 21\)
3. \(8(1 - 9) = 8 \cdot (-8) = -64\)

Now, summing these:

\[
3 + 21 - 64 = -40
\]

Taking the absolute value:

\[
\left| -40 \right| = 40
\]

Finally, computing the area:

\[
\text{Area} = \frac{1}{2} \cdot 40 = 20
\]

Thus, the area of the triangle ABC is \( \boxed{20} \) square units.