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) have 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) have same x-coordinate, so BC is vertical. Length BC = 2 - (-1) = 3.

Step3: Determine angle at B

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

Step1: Identify base and height

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

Step2: Calculate area

Area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 3 \times 3 = 4.5$.

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) (or C + (-3,0) = (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.

Answer:

Right triangle

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