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 A, B, C

Points: A(1,2), B(4,2) (same y-coordinate, horizontal line); B(4,2), C(4,-1) (same x-coordinate, vertical line). So AB ⊥ BC, forming a right angle at B. Three points make a triangle.

Step2: Calculate lengths of AB and BC

Length AB: $|4 - 1| = 3$
Length BC: $|2 - (-1)| = 3$

Step3: Determine shape of triangle ABC

Right angle at B, so right triangle.

Step4: Compute area of triangle ABC

Area formula: $\frac{1}{2} \times base \times height = \frac{1}{2} \times AB \times BC$
Substitute values: $\frac{1}{2} \times 3 \times 3 = 4.5$

Step5: Find coordinates of D for rectangle ABCD

In rectangle, opposite sides equal and parallel. Coordinates of D: (A_x, C_y) = (1, -1) (since A and D share x-coordinate, C and D share y-coordinate).

Answer:

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