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 AB

Points A(1,2) and B(4,2): same y-coordinate, so AB is horizontal. Length AB = 4 - 1 = 3.

Step2: Analyze coordinates of BC

Points B(4,2) and C(4,-1): same x-coordinate, so BC is vertical. Length BC = 2 - (-1) = 3.

Step3: Determine angle at B

Horizontal and vertical lines are perpendicular, so angle at B is 90°. Three points form a triangle with right angle.

Step1: Identify legs of triangle

AB = 3 (horizontal leg), BC = 3 (vertical leg).

Step2: Calculate area

Area of right triangle = $\frac{1}{2} \times \text{leg}_1 \times \text{leg}_2 = \frac{1}{2} \times 3 \times 3 = 4.5$.

Step1: Use rectangle properties

In rectangle ABCD, opposite sides are equal and parallel. Vector AB = (3,0), so vector DC should be (3,0). Vector BC = (0,-3), so vector AD should be (0,-3).

Step2: Find D coordinates

A(1,2) + vector AD = (1,2) + (0,-3) = (1,-1). Verify: D(1,-1), DC = (4-1, -1 - (-1)) = (3,0) = AB; AD = (0,-3) = BC.

Answer:

Right triangle