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:

Sub - Question 1

Step 1: Analyze coordinates of points

Point \( A(1,2) \), \( B(4,2) \), \( C(4, - 1) \). The \( y \) - coordinate of \( A \) and \( B \) is the same (\( y = 2 \)), so \( AB \) is a horizontal line segment. The \( x \) - coordinate of \( B \) and \( C \) is the same (\( x = 4 \)), so \( BC \) is a vertical line segment.

Step 2: Determine the angle between \( AB \) and \( BC \)

Since \( AB \) is horizontal and \( BC \) is vertical, the angle between \( AB \) and \( BC \) is \( 90^{\circ} \). Also, we have three points, so the figure is a right - angled triangle.

Step 1: Calculate the length of \( AB \)

For two points \( (x_1,y_1) \) and \( (x_2,y_2) \) with the same \( y \) - coordinate, the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \) simplifies to \( |x_2 - x_1| \). For \( A(1,2) \) and \( B(4,2) \), \( AB=\vert4 - 1\vert=3 \).

Step 2: Calculate the length of \( BC \)

For two points \( (x_1,y_1) \) and \( (x_2,y_2) \) with the same \( x \) - coordinate, the distance formula simplifies to \( |y_2 - y_1| \). For \( B(4,2) \) and \( C(4,-1) \), \( BC=\vert2-(-1)\vert = 3 \).

Step 3: Calculate the area of the right - angled triangle

The area of a right - angled triangle is given by \( A=\frac{1}{2}\times\text{base}\times\text{height} \). Here, base \( AB = 3 \) and height \( BC = 3 \), so \( A=\frac{1}{2}\times3\times3=\frac{9}{2}=4.5 \).

Step 1: Recall the properties of a rectangle

In a rectangle \( ABCD \), \( AB\parallel CD \) and \( AD\parallel BC \), and \( AB = CD \), \( AD = BC \). Since \( AB \) is horizontal from \( (1,2) \) to \( (4,2) \), \( CD \) should also be horizontal. Since \( BC \) is vertical from \( (4,2) \) to \( (4,-1) \), \( AD \) should also be vertical.

Step 2: Determine the coordinates of \( D \)

The \( x \) - coordinate of \( D \) should be the same as the \( x \) - coordinate of \( A \) (because \( AD \) is vertical), so \( x = 1 \). The \( y \) - coordinate of \( D \) should be the same as the \( y \) - coordinate of \( C \) (because \( CD \) is horizontal), so \( y=-1 \). So the coordinates of \( D \) are \( (1,-1) \).

Step 3: Verify the rectangle

• Check \( AB \) and \( CD \): \( AB \) has length \( 3 \) (from \( x = 1 \) to \( x = 4 \) at \( y = 2 \)), \( CD \) has length \( 3 \) (from \( x = 1 \) to \( x = 4 \) at \( y=-1 \)), and they are both horizontal, so \( AB\parallel CD \) and \( AB = CD \).
• Check \( BC \) and \( AD \): \( BC \) has length \( 3 \) (from \( y = 2 \) to \( y=-1 \) at \( x = 4 \)), \( AD \) has length \( 3 \) (from \( y = 2 \) to \( y=-1 \) at \( x = 1 \)), and they are both vertical, so \( BC\parallel AD \) and \( BC = AD \). Also, all angles are \( 90^{\circ} \) (since \( AB \) is horizontal and \( BC \) is vertical, \( AD \) is vertical and \( CD \) is horizontal).

Answer:

The figure is a right - angled triangle.

Sub - Question 2