Question

Find the area, in square units, of ABC plotted below.

Answer 1

The area, in square units, of triangle
`ABC` is
`52` square units. Therefore, option C is correct

To find the area of triangle
`ABC` with vertices
`\( A(0, 7) \), \( B(7, -2) \),` and
`\( C(-3, -4) \)`, we can use the formula for the area of a triangle given by the coordinates of its vertices:

`\[Area = (1)/(2) |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\]`

where
`\( (x_1, y_1) \)`,
`\( (x_2, y_2) \)`, and
`\( (x_3, y_3) \)` are the coordinates of the vertices
`\( A \)`,
`\( B \)`, and
`\( C \)`, respectively.

Let's calculate it.

To calculate the area of a triangle with vertices at given coordinates, we use the following formula derived from the determinant of a matrix:

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

The coordinates for the vertices
`\( A \), \( B \),` and
`\( C \)` are
`\( A(0, 7) \)`,
`\( B(7, -2) \)`, and
`\( C(-3, -4) \)` respectively. Let's label them as follows:

`\( A = (x_1, y_1) = (0, 7) \). \( B = (x_2, y_2) = (7, -2) \), \( C = (x_3, y_3) = (-3, -4) \)`

Now let's perform the calculation step by step:

1. Calculate the differences in y-coordinates between the vertices :

`\( y_2 - y_3 = -2 - (-4) = -2 + 4 = 2 \)`

`\( y_3 - y_1 = -4 - 7 = -11 \)`

`\( y_1 - y_2 = 7 - (-2) = 7 + 2 = 9 \)`

2. Multiply the x-coordinate of each vertex by the corresponding difference in y-coordinates calculated above:

`\( x_1 \cdot (y_2 - y_3) = 0 \cdot 2 = 0 \) (since \( x_1 = 0 \), this term will be zero)`

`\( x_2 \cdot (y_3 - y_1) = 7 \cdot (-11) = -77 \)`

`\( x_3 \cdot (y_1 - y_2) = -3 \cdot 9 = -27 \)`

3. Sum up the results of the multiplications:

`( Sum = 0 + (-77) + (-27) = -104 \)`

4. Take the absolute value and divide by 2 to get the area:

`\( Area = (1)/(2) \cdot |-104| = (1)/(2) \cdot 104 = 52 \)`

So, the step-by-step detailed calculation confirms that the area of triangle
`ABC` is
`52` square units.