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 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 verif the rectangle by checking the properties of its sides.

Explanation:

Step1: Analyze side relationships

Points $A(1,2)$, $B(4,2)$: $y$-coordinates are equal, so $AB$ is horizontal. Length $AB = 4-1 = 3$.
Points $B(4,2)$, $C(4,-1)$: $x$-coordinates are equal, so $BC$ is vertical. Length $BC = 2-(-1) = 3$.
$AB$ is perpendicular to $BC$, and we have 3 connected points, so the shape is a right triangle.

Step2: Calculate triangle area

Area of a right triangle is $\frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$.
Substitute $AB=3$, $BC=3$:
$\text{Area} = \frac{1}{2} \times 3 \times 3$

Step3: Find coordinates of point D

For $ABCD$ to be a rectangle, opposite sides are equal and parallel. $AD$ must match $BC$ (vertical, length 3), and $CD$ must match $AB$ (horizontal, length 3).
From $A(1,2)$, move down 3 units (same direction as $BC$): $y=2-3=-1$, $x$ stays 1. So $D(1,-1)$.
Verify: $AD$ is vertical ($x=1$ for $A$ and $D$), length $2-(-1)=3=BC$; $CD$ is horizontal ($y=-1$ for $C$ and $D$), length $4-1=3=AB$. All angles are right angles.

Answer:

1. The figure is a right triangle.
2. $\frac{9}{2}$ (or 4.5)
3. The coordinates of point $D$ are $(1, -1)$. When plotted, $AD$ is vertical (matches the vertical side $BC$) and $CD$ is horizontal (matches the horizontal side $AB$), with all angles being right angles, confirming $ABCD$ is a rectangle.