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: $4-1=3$.

Step2: Analyze coordinates of BC

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

Step3: Determine angle at B

Horizontal and vertical lines are perpendicular, so $\angle ABC=90^\circ$. Three points form a triangle with right angle at B.

Step4: Calculate area of triangle

Area formula: $\frac{1}{2} \times base \times height$. Base AB=3, height BC=3. Area: $\frac{1}{2} \times 3 \times 3 = 4.5$.

Step5: Find coordinates of D for rectangle

In rectangle ABCD, opposite sides are equal and parallel. A to B is (3,0), so C to D should be (3,0): D=C+(3,0)? No, A to D should equal B to C. B to C is (0,-3), so D=A+(0,-3)=(1,2-3)=(1,-1).

Answer:

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