Question

Plot each point and form the triangle \(A B C\). Show that the triangle is a right triangle. Find its area. $$ A=(4,-3) ; \quad B=(0,-3) ; \quad C=(4,2) $$

Short answer

Plot points and find the length of formed triangle. Verify its rightness, calculate area.

Step-by-step solution

Plot the Points

Plot the points A(4, -3), B(0, -3), and C(4, 2) on a coordinate plane.

Form the Triangle

Connect the points A, B, and C to form triangle ABC.

Calculate the Lengths of the Sides

Use the distance formula to find the lengths of sides AB, BC, and AC:d_{AB} = x_2 - x_1 = 4 - 0 = 4

Key concepts

Plot Points

To start with, you need to plot the points given in the exercise, which are A(4, -3), B(0, -3), and C(4, 2). Plotting points involves placing the points on a coordinate plane based on their coordinates.

Here’s how:
- The first number in the pair (4, -3) is the x-coordinate, and it tells you how far to move horizontally from the origin (0, 0).
- The second number in the pair represents the y-coordinate and tells you how far to move vertically.

So for A(4, -3), move 4 units to the right and 3 units down. For B(0, -3), stay at the origin and move 3 units down. For C(4, 2), move 4 units right and 2 units up. By consistently plotting points this way, the rest of the problem becomes more straightforward.

Distance Formula

After plotting the points, we need to calculate the lengths of the sides of the triangle using the distance formula.
The distance formula helps to find the distance between two points on a coordinate plane and is given by:
\[d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\]
For side AB, (x1, y1) are the coordinates of point A, and (x2, y2) are the coordinates of point B.
By calculating:

\[d_{AB} = \sqrt{{(0 - 4)^2 + (-3 - (-3))^2}} = 4\] \[d_{BC} = \sqrt{{(4 - 0)^2 + (-3 - 2)^2}} = 5\] \[d_{AC} = \sqrt{{(4 - 4)^2 + (2 - (-3))^2}} = 5\]

We recognize the distances, confirming the sides AB, BC, and AC. Using the distance formula ensures that even if you make mistakes in drawing the triangle, the math will set you right.

Coordinate Plane

The coordinate plane is a fundamental concept in geometry that allows us to visually represent algebraic equations and geometrical shapes, like triangles.
It is a two-dimensional surface formed by two perpendicular lines called axes:

- The horizontal line is the x-axis.
- The vertical line is the y-axis.
- These axes intersect at a point called the origin (0,0).
- Positive x and y values go right and upward, while negative values go left and downward, respectively.

Understanding this helps in plotting points and analyzing the shapes created by them. It provides a clear way to see how formulas and equations work out in a real-world setting.

Triangle Area

Finding the area of a triangle is yet another important aspect. Specifically, for right triangles, there’s a simplified formula to find the area:
\[area = \frac{1}{2} (base \times height)\]
In this case, since we have a right triangle with the base on the x-axis (between points B and A) and the height along the y-axis (between B and C),
the formula becomes:
\[area = \frac{1}{2} (AB \times BC)\ = \frac{1}{2} (4 \times 5) = 10\]
You can always work with the lengths computed with the distance formula to confirm the area is correct.
This simplified method is specific to right triangles because the base and height are the legs of the triangle.