Completing Grid Figures
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ISEE Lower Level: Mathematics Achievement › Completing Grid Figures
A horizontal line segment starts at (3, 4) and ends at (9, 4). A vertical line segment starts at (3, 4) and ends at (3, 8). These are two sides of a rectangle. What is the fourth vertex of this rectangle?
(8, 9)
(9, 8)
(6, 6)
(9, 4)
Explanation
The problem describes three vertices of the rectangle: the shared point (3, 4), the end of the horizontal segment (9, 4), and the end of the vertical segment (3, 8). The fourth vertex must align horizontally with (3, 8) (so its y-coordinate is 8) and vertically with (9, 4) (so its x-coordinate is 9). Therefore, the fourth vertex is (9, 8).
The points A(2, 2) and B(5, 2) form the bottom side of a square. What are the coordinates of the vertex that is directly above point B?
(5, 5)
(2, 5)
(5, -1)
(8, 2)
Explanation
The side AB is horizontal and has a length of 5 - 2 = 3 units. Since it is the bottom side of a square, the other two vertices must be 3 units directly above points A and B. The vertex directly above B(5, 2) will have the same x-coordinate (5) and a y-coordinate that is 3 units greater (2 + 3 = 5). Thus, the coordinates are (5, 5).
Three corners of a square are located at (-1, -1), (3, -1), and (3, 3). What are the coordinates of the fourth corner?
(1, -3)
(-1, -5)
(-1, 3)
(7, 3)
Explanation
The side from (-1, -1) to (3, -1) is horizontal and has length 4. The side from (3, -1) to (3, 3) is vertical and has length 4. To complete the square, the fourth corner must have the x-coordinate of (-1, -1) and the y-coordinate of (3, 3). Therefore, the fourth vertex is at (-1, 3).
Points (-3, 2), (4, 2), and (4, -1) are three vertices of a rectangle. What is the fourth vertex?
(11, -1)
(-3, -1)
(-3, 5)
(-1, -3)
Explanation
The side between (-3, 2) and (4, 2) is horizontal. The side between (4, 2) and (4, -1) is vertical. To complete the rectangle, the fourth vertex must align vertically with (-3, 2) (having an x-coordinate of -3) and horizontally with (4, -1) (having a y-coordinate of -1). Therefore, the fourth vertex is at (-3, -1).
A rectangle has vertices at (1, 2), (7, 2), and (7, 5). What are the coordinates of the fourth vertex?
(1, 5)
(5, 1)
(1, -1)
(13, 5)
Explanation
The side from (1, 2) to (7, 2) is horizontal. The side from (7, 2) to (7, 5) is vertical. To form a rectangle, the fourth vertex must align horizontally with (7, 5) and vertically with (1, 2). This means it shares the x-coordinate of (1, 2) and the y-coordinate of (7, 5). The fourth vertex is (1, 5).
Three vertices of a square are at the points (2, 1), (5, 1), and (5, 4). What are the coordinates of the fourth vertex?
(2, -2)
(4, 2)
(2, 4)
(8, 4)
Explanation
The vertices (2, 1) and (5, 1) form a horizontal side of length 3. The vertices (5, 1) and (5, 4) form a vertical side of length 3. To complete the square, the fourth vertex must have the x-coordinate of (2, 1) and the y-coordinate of (5, 4). Therefore, the fourth vertex is at (2, 4).