A rectangle is drawn on a coordinate grid. Three of its four vertices are located at points $(-1,-2),(-1,4)$, and $(2,-2)$. What is the location of the fourth vertex?
Answer (2)
Analyze the coordinates of the given vertices to determine the orientation and lengths of the rectangle's sides.
Deduce that the missing vertex must form sides parallel to the existing ones.
Determine the coordinates of the fourth vertex by finding the intersection of lines parallel to the sides.
Conclude that the fourth vertex is located at ( 2 , 4 ) .
Explanation
Analyze the given points and determine the sides of the rectangle. Let's analyze the given points to determine the sides of the rectangle. We have three vertices: A ( − 1 , − 2 ) , B ( − 1 , 4 ) , and C ( 2 , − 2 ) .
Since points A and B have the same x-coordinate, the segment A B is vertical. The length of A B is ∣4 − ( − 2 ) ∣ = 6 .
Since points A and C have the same y-coordinate, the segment A C is horizontal. The length of A C is ∣2 − ( − 1 ) ∣ = 3 .
Therefore, A B and A C are perpendicular to each other, and they are adjacent sides of the rectangle. Let the fourth vertex be D ( x , y ) . Since A BC D is a rectangle, A D must be parallel to BC , and C D must be parallel to A B .
Since C D is parallel to A B , C D is vertical, so the x-coordinate of D must be the same as the x-coordinate of C , which is 2. Since A D is parallel to BC , A D is horizontal, so the y-coordinate of D must be the same as the y-coordinate of B , which is 4. Therefore, the fourth vertex is D ( 2 , 4 ) .
Verify the location of the fourth vertex. To verify, we can check that B D is parallel to A C . Since A C is horizontal, B D must also be horizontal. The y-coordinate of B is 4, and the y-coordinate of D is 4, so B D is horizontal. Also, B D = ∣2 − ( − 1 ) ∣ = 3 , which is equal to A C .
We can also check that C D is parallel to A B . Since A B is vertical, C D must also be vertical. The x-coordinate of C is 2, and the x-coordinate of D is 2, so C D is vertical. Also, C D = ∣4 − ( − 2 ) ∣ = 6 , which is equal to A B .
State the final answer. The location of the fourth vertex is ( 2 , 4 ) .
Examples
Understanding coordinate geometry is useful in many real-world applications, such as mapping and navigation. For example, GPS systems use coordinates to locate positions on Earth. Architects and engineers use coordinate systems to design buildings and structures. In computer graphics, coordinate systems are used to create and manipulate images and animations. This problem demonstrates how to use the properties of rectangles and coordinate geometry to find the location of an unknown point, which is a fundamental skill in these fields.
The fourth vertex of the rectangle is located at the coordinates ( 2 , 4 ) . This point maintains the rectangle's properties by aligning with the existing side lengths and orientations. Verifying the arrangements of the vertices confirms this location as accurate.
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