Final answer:
To determine whether the given points form a rhombus, square, or rectangle, we can calculate the distances between the points and check the slopes. The given points form a rhombus, square, and rectangle.
Step-by-step explanation:
To determine whether the given points form a rhombus, square, or rectangle, we need to analyze the properties of each shape.
A rhombus is a quadrilateral with all sides having the same length. To check this, we can calculate the distances between the points.
Using the distance formula, the lengths of AB, BC, CD, and DA are:
AB = sqrt((4-5)^2 + (10-10)^2) = 1
BC = sqrt((4-4)^2 + (9-10)^2) = 1
CD = sqrt((5-4)^2 + (9-9)^2) = 1
DA = sqrt((5-5)^2 + (10-9)^2) = 1
Since all the sides are equal, the given points form a rhombus.
A square is a special case of a rhombus where all angles are right angles. We can check this by calculating the slopes of consecutive sides.
The slope of AB = (10-10) / (4-5) = 0
The slope of BC = (9-10) / (4-4) = -1
The slope of CD = (9-9) / (5-4) = 0
The slope of DA = (10-9) / (5-5) = 1
Since all the slopes are 0 or 1, the given points form a square.
A rectangle is a quadrilateral with opposite sides having equal lengths and all angles being right angles. We have already calculated the lengths of the sides, and they are equal. To check if the angles are right angles, we can calculate the slopes of adjacent sides.
The slope of AB = 0
The slope of BC = -1
The slope of CD = 0
The slope of DA = 1
Since all the slopes are 0 or 1, the given points form a rectangle.