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Shape and Space - Coordinate Geometry in the First Quadrant

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Coordinate Plane: In the first quadrant, we use a grid formed by two perpendicular lines. The horizontal line is called the xx-axis and the vertical line is called the yy-axis. They intersect at a 90∘90^{\circ} angle. Visually, imagine an 'L' shaped frame where all movements are to the right and upwards.

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The Origin: This is the starting point of the coordinate system, located at the intersection of the two axes. Its coordinates are always (0,0)(0, 0). In a visual representation, it is the bottom-left corner of the grid.

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Ordered Pairs (x,y)(x, y): Every point on the grid is identified by a pair of numbers called coordinates. The first number (xx) tells you the horizontal distance from the origin, and the second number (yy) tells you the vertical distance. A common memory trick is: 'You must go along the hallway (xx) before you go up the stairs (yy)'.

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Plotting Points: To plot a point such as (5,3)(5, 3), you start at the origin (0,0)(0, 0), move 55 units to the right along the xx-axis, and then move 33 units straight up. The point is visually marked where the vertical line from x=5x=5 meets the horizontal line from y=3y=3.

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Axes and Scales: The axes are numbered lines. The scale tells you the value of each square on the grid. While many grids use a scale of 11 (counting 1,2,3...1, 2, 3...), some grids might use scales of 2,52, 5, or 1010. Always check the labels on the xx and yy lines to determine the value of each grid unit.

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Translation (Shifting): Translation involves sliding a point or shape to a new position without changing its size or orientation. Visually, if you move a point '4 units right and 2 units up', its xx-coordinate increases by 44 and its yy-coordinate increases by 22.

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Horizontal and Vertical Lines: If multiple points have the same yy-coordinate, they form a horizontal line. If they have the same xx-coordinate, they form a vertical line. For example, points (2,5)(2, 5), (4,5)(4, 5), and (7,5)(7, 5) all lie on a flat horizontal line 55 units above the xx-axis.

📐Formulae

Coordinate Notation: (x,y)(x, y)

Translation (Right/Up): (x+a,y+b)(x + a, y + b)

Translation (Left/Down): (x−a,y−b)(x - a, y - b)

Horizontal Distance (same yy): d=x2−x1d = x_{2} - x_{1}

Vertical Distance (same xx): d=y2−y1d = y_{2} - y_{1}

💡Examples

Problem 1:

Point AA is located at (2,4)(2, 4). If Point AA is translated 55 units to the right and 33 units up to create Point BB, what are the coordinates of Point BB?

Solution:

  1. Identify the starting coordinates: x=2,y=4x = 2, y = 4.
  2. Apply the horizontal translation (right): xnew=2+5=7x_{new} = 2 + 5 = 7.
  3. Apply the vertical translation (up): ynew=4+3=7y_{new} = 4 + 3 = 7.
  4. Write the new ordered pair: (7,7)(7, 7).

Explanation:

To move right, we add to the xx-coordinate. To move up, we add to the yy-coordinate. The point slides diagonally across the grid from (2,4)(2, 4) to (7,7)(7, 7).

Problem 2:

A rectangle has four vertices. Three of the vertices are at (1,1)(1, 1), (5,1)(5, 1), and (5,4)(5, 4). What is the coordinate of the fourth vertex to complete the rectangle?

Solution:

  1. Look at the xx-coordinates: We have points at x=1x = 1 and x=5x = 5.
  2. Look at the yy-coordinates: We have points at y=1y = 1 and y=4y = 4.
  3. The missing point must share the xx-coordinate of the first point (11) and the yy-coordinate of the third point (44) to close the shape.
  4. The fourth vertex is (1,4)(1, 4).

Explanation:

In a rectangle on a coordinate plane, pairs of vertices must share the same xx or yy values to create perfectly horizontal and vertical sides.