Which of the following is the fourth vertex needed to create a rectangle with vertices located at (5, 6), (9, 6), and (5, 3)?
A. (5, -6)
B. (5, -3)
C. (9, -6)
D. (9, 3)
Answer (1)
Analyze the coordinates of the given vertices.
Recognize that the sides of the rectangle are parallel to the x and y axes.
Determine the coordinates of the fourth vertex based on the properties of a rectangle.
The fourth vertex is ( 9 , 3 ) .
Explanation
Analyze the given vertices Let's analyze the given vertices of the rectangle: (5, 6), (9, 6), and (5, 3). We need to find the fourth vertex.
Determine the coordinates of the fourth vertex Let the given vertices be A(5, 6), B(9, 6), and C(5, 3). Since it's a rectangle, the sides are parallel to the x and y axes. The side AB is parallel to the x-axis because the y-coordinates of A and B are the same. The side AC is parallel to the y-axis because the x-coordinates of A and C are the same. Therefore, AB and AC are adjacent sides of the rectangle. Let the fourth vertex be D(x, y). Since ABCD is a rectangle, AD is parallel to BC and CD is parallel to AB. Since CD is parallel to AB, the y-coordinate of D is the same as the y-coordinate of C, so y = 3. Since AD is parallel to BC, the x-coordinate of D is the same as the x-coordinate of B, so x = 9. Therefore, the fourth vertex is D(9, 3).
State the final answer The fourth vertex needed to create a rectangle with vertices located at (5, 6), (9, 6), and (5, 3) is (9, 3).
Examples
Understanding coordinate geometry is crucial in various fields, such as mapping and urban planning. For instance, when designing a city block, architects use coordinates to define the boundaries of buildings and parks, ensuring precise alignment and efficient use of space. This problem demonstrates how to find the missing vertex of a rectangle, which is a fundamental concept in ensuring accurate spatial arrangements in real-world applications.
