# When the coordinates (1, 1), (2, 3), (5, 3), and (4, 1) are joined, which shape is formed?

Coordinate geometry is used to represent various equations as curves on the cartesian plane. It has many applications in engineering which include building trajectories of rockets and many others.

## Answer: When the coordinates (1, 1), (2, 3), (5, 3), and (4, 1) are joined, the shape formed is a parallelogram.

Let's understand the solution in detail.

**Explanation:**

First of all, we calculate the distances between two points.

Distance between (1, 1) and (2, 3) = √{(2 - 1)^{2} + (3 - 1)^{2}} = √5 units

Distance between (2, 3) and (5, 3) = √{(5 - 2)^{2} + (3 - 3)^{2}} = 3 units

Distance between (5, 3) and (4, 1) = √{(4 - 5)^{2} + (1 - 3)^{2}} = √5 units

Distance between (4, 1) and (1, 1) = √{(4 - 1)^{2} + (1 - 1)^{2}} = 3 units

Here, we see that two pairs of line segments are equal.

Now, we find the slopes of all the curves using the slope formula.

Slope of line joining (1, 1) and (2, 3) = (3 - 1) / (2 - 1) = 2

Slope of line joining (2, 3) and (5, 3) = (3 - 3) / (5 - 2) = 0

Slope of line joining (5, 3) and (4, 1) = (3 - 1) / (5 - 4) = 2

Slope of line joining (1, 1) and (4, 1) = (1 - 1) / (4 - 1) = 0

Hence, we see that two pairs of slopes are equal. Hence, two pairs of lines are parallel in the shape formed.

Therefore, we can say that the shape formed is a parallelogram since two pairs of lines are equal and parallel. The shape is shown in the graph below.