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 side AB

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

Step2: Analyze side BC

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

Step3: Determine angle at B

Horizontal and vertical lines form a right angle. Three points form a triangle with right angle at B.

Step1: Identify base and height

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

Step2: Calculate area

Area formula for triangle: $\frac{1}{2} \times base \times height$.
$\frac{1}{2} \times 3 \times 3 = 4.5$.

Step1: Find coordinates of D

In rectangle ABCD, opposite sides are 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 D = A + (0,-3) = (1,-1).

Step2: Verify rectangle properties

AD: from (1,2) to (1,-1) (vertical, length 3, same as BC).
CD: from (4,-1) to (1,-1) (horizontal, length 3, same as AB).
All angles are right angles (horizontal/vertical sides).

Answer:

Right triangle