Question

question:
in the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, - 1) respectively.

1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure?
2. calculate the area of the figure formed by points a, b, and c.
3. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

Explanation:

Step1: Analyze coordinates of A, B, C

A(1,2), B(4,2): same y-coordinate (horizontal line). B(4,2), C(4,-1): same x-coordinate (vertical line). So AB ⊥ BC. Three points form a triangle with right angle at B.

Step2: Calculate lengths of AB and BC

AB length: $|4 - 1| = 3$. BC length: $|2 - (-1)| = 3$. Area of right triangle: $\frac{1}{2} \times AB \times BC = \frac{1}{2} \times 3 \times 3 = 4.5$.

Step3: Find D for rectangle ABCD

In rectangle, opposite sides equal and parallel. A(1,2) to B(4,2) is +3 in x; B(4,2) to C(4,-1) is -3 in y. So C(4,-1) to D should be -3 in x: $4 - 3 = 1$, y stays: -1. So D(1,-1). Verify: A(1,2) to D(1,-1) is vertical, D(1,-1) to C(4,-1) is horizontal, right angles exist.

Answer:

1. Right triangle
2. 4.5
3. (1, -1)