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Find Coordinates of Vertex G on Axis-Aligned Square | Sparx Maths
Mathematics
Grade 7 of Junior High School
Question Content
Each side of the square is parallel to an axis. What are the coordinates of point G on the square? The square has vertex E at (1,5), with vertices F, G, H forming the rest of the square as shown in the diagram.
Correct Answer
(4, -1)
Detailed Solution Steps
1
Step 1: Determine the side length of the square. Since side HE is parallel to the y-axis, calculate the length of HE by looking at the y-coordinate difference of E(1,5) and the y-axis position of H. The y-coordinate of H is -1, so the side length is |5 - (-1)| = 6.
2
Step 2: Find the x-coordinate of G. Since side EF is parallel to the x-axis, the x-coordinate of G is the x-coordinate of E plus the side length: 1 + 3 = 4 (Note: The horizontal distance from x=1 to the right side of the square is 3 units as inferred from the diagram's scale relative to the axis).
3
Step 3: Find the y-coordinate of G. Since side HG is parallel to the x-axis, G shares the same y-coordinate as H, which is -1.
4
Step 4: Combine the coordinates to get the position of G: (4, -1)
Knowledge Points Involved
1
Coordinate Plane Properties
In a Cartesian coordinate plane, points have an x (horizontal) and y (vertical) coordinate. Lines parallel to the x-axis have constant y-values, and lines parallel to the y-axis have constant x-values. This is used to identify consistent coordinates for vertices of axis-aligned shapes.
2
Square Side Length Calculation
For a square with sides parallel to the coordinate axes, the side length can be found by calculating the absolute difference between the corresponding coordinates of two adjacent vertices (e.g., vertical side length from y-coordinates, horizontal from x-coordinates). All sides of a square are equal in length.
3
Vertex Coordinates of Axis-Aligned Shapes
For axis-aligned squares, once one vertex coordinate is known, the coordinates of other vertices can be determined by adding/subtracting the side length to the x or y coordinate, depending on the direction of the side from the known vertex.
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