In the Cartesian coordinate plane shown, plot the points $A(2{,}3)$, $B(-2,-1)$, and $C(2,-3)$, and connect them in order $A\to B\to C$. Then find the area of $\triangle ABC$.

Step 1: Plot the points and draw the triangle

First, we plot the given points $A(2{,}3)$, $B(-2,-1)$, and $C(2,-3)$ on the Cartesian coordinate plane. Then, we connect these points in the specified order to form triangle $ABC$.

Step 2: Identify the base and height of the triangle

Notice that points $A(2{,}3)$ and $C(2,-3)$ have the same $x$ -coordinate. This means the side $AC$ is a vertical line segment, which can serve as the base of the triangle. The length of the base $AC$ is the absolute difference of their $y$ -coordinates. The height of the triangle with respect to this base will be the perpendicular distance from point $B$ to the line containing $AC$, which is the absolute difference of the $x$ -coordinates of $A$ (or $C$) and $B$.

Step 3: Calculate the length of the base

Using the coordinates of points $A(2{,}3)$ and $C(2,-3)$, we calculate the length of the base $AC$. The length is the absolute difference between their $y$ -coordinates.

Step 4: Calculate the height of the triangle

The height of the triangle is the perpendicular distance from point $B(-2,-1)$ to the line segment $AC$. Since $AC$ lies on the line $x=2$, the height is the absolute difference between the $x$ -coordinate of $B$ and the $x$ -coordinate of $AC$.

Step 5: Calculate the area of the triangle

Now we use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. We substitute the calculated base length of 6 units and height of 4 units into the formula.

Step 6: Final Calculation

Performing the multiplication, we find the area of $\triangle ABC$ to be 12 square units.

More from this chapter

• The x-coordinate of the point which lies on the line represented by 3x - y - 1 = 0 and whose y-coordinate is 5, is:
• Find the distance between the points (1, 2) and (4, 6).
• Show that the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square.
• The distance of ( - 4, 7) from y-axis is:
• Find the equations of the lines parallel to axes and passing through ( -2, 3).
• Find the distance between the points (3, 4) and (-3, -4).

All Orienting Yourself: The Use of Coordinates solutions →