Three vertices of a rectangle are A(−1,5) , B(4,5) , and C(4,−2) . Find the fourth vertex, D
To find the fourth vertex of the rectangle, we can use the fact that opposite sides of a rectangle are parallel and equal in length.
We know that AB is one of the sides of the rectangle and BC is another side. Since AB is horizontal, the side AD must also be horizontal and parallel to AB. This means that the y-coordinate of D is the same as the y-coordinate of A, which is 5.
Similarly, since BC is vertical, the side CD must also be vertical and parallel to BC. This means that the x-coordinate of D is the same as the x-coordinate of C, which is 4.
Therefore, the fourth vertex of the rectangle is D(4, 5).
wrong
Apologies for the mistake.
Let's find the length of AB and BC.
Distance between A and B:
√((4 - (-1))^2 + (5 - 5)^2) = √(5^2 + 0^2) = 5 units
Distance between B and C:
√((4 - 4)^2 + (-2 - 5)^2) = √(0^2 + (-7)^2) = 7 units
Since opposite sides of a rectangle are equal in length, the length of AD should also be 5 units and CD should also be 7 units.
Since AD is perpendicular to AB, its equation will be x = -1.
Since BC is perpendicular to CD, its equation will be y = -2.
Therefore, the fourth vertex D will have the coordinates (-1, -2).
Given the vertices A(−2,4) , B(1,4) , and C(1,1) , find the other vertex that is needed to construct the square
To construct a square, we need two adjacent sides to be perpendicular and equal in length.
Let's calculate the distance between the given vertices to determine the side length of the square:
Distance between A and B:
√((1 - (-2))^2 + (4 - 4)^2) = √(3^2 + 0^2) = 3 units
Distance between B and C:
√((1 - 1)^2 + (1 - 4)^2) = √(0^2 + (-3)^2) = 3 units
Since the distances between A and B and between B and C are equal, we can conclude that AB and BC are the adjacent sides we need to construct a square.
To find the other vertex, we can use the midpoint formula. The midpoint of AB is the same as the midpoint of CD, so we can obtain the coordinates of D by finding the midpoint of BC.
Midpoint of BC:
x-coordinate: (1 + 1) / 2 = 2 / 2 = 1
y-coordinate: (4 + 1) / 2 = 5 / 2 = 2.5
Therefore, the other vertex of the square is D(1, 2.5).
not right
Apologies for the earlier mistake. Let's find the length of AB and BC.
Distance between A and B:
√((1 - (-2))^2 + (4 - 4)^2) = √(3^2 + 0^2) = √(9) = 3 units
Distance between B and C:
√((1 - 1)^2 + (1 - 4)^2) = √(0^2 + (-3)^2) = √(9) = 3 units
Since the distances between A and B and between B and C are equal, we can conclude that AB and BC are the adjacent sides we need to construct a square.
To find the other vertex, we can determine the vector from A to B and add it to the coordinates of C.
x-coordinate of D: 1 + (4 - (-2)) = 1 + 6 = 7
y-coordinate of D: 1 + (4 - 4) = 1 + 0 = 1
Therefore, the other vertex of the square is D(7, 1).