Question

Triangle ABC has vertices as shown. What is the area of the triangle?
graph shows 3 points a(1,4) b(4,1) c(-3,1)


(1 point)
Responses

21 square units
21 square units

10.5 square units
10.5 square units

27–√
square units
2 square root of 7 square units

15−−√
square units

Answer 1

To calculate the area of triangle ABC with vertices A(1,4), B(4,1), and C(-3,1), we can use the formula for the area of a triangle given its vertices (x1, y1), (x2, y2), (x3, y3):

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

Plugging in the coordinates of the points A(1,4), B(4,1), and C(-3,1):

- \( x_1 = 1, y_1 = 4 \)
- \( x_2 = 4, y_2 = 1 \)
- \( x_3 = -3, y_3 = 1 \)

Now substituting these values into the area formula:

\[
\text{Area} = \frac{1}{2} \left| 1(1 - 1) + 4(1 - 4) + (-3)(4 - 1) \right|
\]

Calculating each term:

\[
= \frac{1}{2} \left| 1(0) + 4(-3) + (-3)(3) \right|
= \frac{1}{2} \left| 0 - 12 - 9 \right|
= \frac{1}{2} \left| -21 \right|
= \frac{1}{2} \times 21
= 10.5
\]

Therefore, the area of triangle ABC is **10.5 square units**.