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 lengths/angles

First, calculate distances between points:

• $AB$: $\sqrt{(4-1)^2+(2-2)^2} = 3$
• $BC$: $\sqrt{(4-4)^2+(-1-2)^2} = 3$
• $AC$: $\sqrt{(4-1)^2+(-1-2)^2} = 3\sqrt{2}$

Check if $AB^2 + BC^2 = AC^2$: $3^2+3^2=18=(3\sqrt{2})^2$, so $\angle B=90^\circ$.

Step2: Calculate triangle area

Use right triangle area formula: $\frac{1}{2} \times base \times height$
Expression: $\frac{1}{2} \times AB \times BC = \frac{1}{2} \times 3 \times 3$

Step3: Find coordinates of D

In rectangle $ABCD$, opposite sides are equal. $D$ must match $A$'s x-coordinate and $C$'s y-coordinate.
Expression: $D=(1, -1)$
Verify: $AD=\sqrt{(1-1)^2+(-1-2)^2}=3$, $CD=\sqrt{(1-4)^2+(-1+1)^2}=3$. All angles are 90° as sides are axis-aligned.

Answer:

1. The figure is a right isosceles triangle.
2. $\frac{9}{2}$ (or 4.5)
3. Coordinates of $D$: $(1, -1)$. When plotted, $AB$ and $CD$ are horizontal (length 3), $BC$ and $AD$ are vertical (length 3), so all angles are right angles, confirming a rectangle.