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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 First Quadrant is the top-right section of the coordinate plane where both xx (horizontal) and yy (vertical) values are positive. The origin (0,0)(0, 0) is the starting point.

The First Quadrant coordinate system showing X and Y axes starting from (0,0).
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Coordinates are written as an ordered pair (x,y)(x, y). The first number tells you how many units to move along the horizontal axis, and the second number tells you how many units to move up the vertical axis.

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Distance between points can be calculated by counting grid units. If two points have the same yy-coordinate, the distance is the difference in xx-coordinates: ∣x2−x1∣|x_2 - x_1|. If they have the same xx-coordinate, it is ∣y2−y1∣|y_2 - y_1|.

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A translation moves a shape or point. Adding to xx moves it right, subtracting from xx moves it left. Adding to yy moves it up, subtracting from yy moves it down.

📐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.

Problem 3:

Calculate the length of the horizontal line segment connecting point P(2,6)P(2, 6) and point Q(7,6)Q(7, 6).

A horizontal line segment plotted from (2,6) to (7,6) on a coordinate grid.

Solution:

7−2=5 units7 - 2 = 5 \text{ units}

Explanation:

Since the yy-coordinates are both 66, the line is perfectly horizontal. To find the length, we subtract the smaller xx-coordinate from the larger xx-coordinate: 7−2=57 - 2 = 5.

Problem 4:

A triangle has vertices at A(2,1)A(2, 1), B(5,1)B(5, 1), and C(2,4)C(2, 4). If the triangle is translated 22 units right and 33 units up, what are the new coordinates of vertex C′C'?

A triangle moving from its original position to a new position 2 units right and 3 units up.

Solution:

C′=(2+2,4+3)=(4,7)C' = (2 + 2, 4 + 3) = (4, 7)

Explanation:

To translate a point right, we add to the xx-coordinate (2+2=42 + 2 = 4). To translate it up, we add to the yy-coordinate (4+3=74 + 3 = 7).