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 AB's direction/length

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

Step2: Analyze BC's direction/length

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

Step3: Determine angle at B

AB is horizontal, BC is vertical → angle ABC = 90°. Three points form triangle with right angle.

Step1: Identify base and height

AB = 3 (base), BC = 3 (height) (right angle between them).

Step2: Calculate area

Area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 3 \times 3$.

Step1: Find D for rectangle

In rectangle ABCD, opposite sides equal/parallel. A(1,2) to B(4,2) is +3x; B(4,2) to C(4,-1) is -3y. So D = A + (-3y) = (1,2-3) = (1,-1).

Step2: Verify sides

AD: (1,2) to (1,-1) (vertical, length 3); DC: (1,-1) to (4,-1) (horizontal, length 3). All angles 90°.

Answer:

Right triangle