Given the set of vertices, determine whether parallelogram ABCD is a rhombus, a rectangle, or a square.
List all that apply.
A(5, 10), B(4, 10), C(4, 9), D(5, 9)
a square, with an edge length of one unit
a square is a subset of all the other figures
it is a parallelogram, rhombus, and rectangle
To determine whether parallelogram ABCD is a rhombus, a rectangle, or a square, we need to analyze the properties of each shape.
1. Rhombus:
A rhombus is a quadrilateral with all sides of equal length. To check if ABCD is a rhombus, we need to calculate the lengths of all sides and compare them.
Using the distance formula between two points (x1, y1) and (x2, y2):
AB = √((x2 - x1)^2 + (y2 - y1)^2)
AB = √((4 - 5)^2 + (10 - 10)^2) = √((-1)^2 + 0^2) = √(1 + 0) = 1
BC = √((x2 - x1)^2 + (y2 - y1)^2)
BC = √((4 - 4)^2 + (9 - 10)^2) = √(0^2 + (-1)^2) = √(0 + 1) = 1
CD = √((x2 - x1)^2 + (y2 - y1)^2)
CD = √((5 - 4)^2 + (9 - 9)^2) = √(1^2 + 0^2) = √(1 + 0) = 1
DA = √((x2 - x1)^2 + (y2 - y1)^2)
DA = √((5 - 5)^2 + (9 - 10)^2) = √(0^2 + (-1)^2) = √(0 + 1) = 1
Since AB = BC = CD = DA = 1, all sides have equal length. Thus, ABCD is a rhombus.
2. Rectangle:
A rectangle is a quadrilateral with all angles equaling 90 degrees (right angles). To check if ABCD is a rectangle, we need to calculate the slopes of the opposite sides and check if they are perpendicular to each other.
The slope between two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Slope of line AB:
m_AB = (10 - 10) / (4 - 5) = 0 / (-1) = 0
Slope of line BC:
m_BC = (9 - 10) / (4 - 4) = (-1) / 0 (undefined)
Slope of line CD:
m_CD = (9 - 9) / (5 - 4) = 0 / 1 = 0
Slope of line DA:
m_DA = (9 - 10) / (5 - 5) = (-1) / 0 (undefined)
Since lines BC and DA have undefined slopes, they are vertical, and lines AB and CD have slopes of 0 (horizontal). This indicates that ABCD's opposite sides are parallel and perpendicular, respectively, making it a rectangle.
3. Square:
A square is a special type of rectangle with all sides of equal length and all angles equaling 90 degrees. If we have already determined that ABCD is a rectangle and all sides are equal, it is also a square.
Therefore, based on the analysis, parallelogram ABCD is both a rhombus and a rectangle.