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Geometry - Coordinates and Grid References

Grade 4IB

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

🔑Concepts

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The coordinate plane is a 2D surface defined by two perpendicular lines: the horizontal xx-axis and the vertical yy-axis. The point where they intersect is called the origin, written as (0,0)(0, 0). Each point is identified by an ordered pair (x,y)(x, y). To plot a point, always move along the xx-axis first, then up along the yy-axis.

A coordinate grid showing point P at (3, 2) reached by moving 3 units right and 2 units up.
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Grid references are often used on maps to identify squares instead of exact points. In a grid system, letters usually denote columns (horizontal) and numbers denote rows (vertical). For example, a square might be labeled B3B3.

A 2x2 grid with columns A and B and rows 1 and 2, showing an object in square B2.
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Translation means sliding a shape or point to a new position without rotating it or changing its size. To translate a point, you add or subtract from its coordinates. Moving left/right affects the xx-coordinate; moving up/down affects the yy-coordinate.

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The four quadrants: Grade 4 usually focuses on the first quadrant where both xx and yy are positive. In this area, we always work within the space to the right of and above the origin.

📐Formulae

Coordinate Format: (x,y)(x, y) where xx = horizontal distance and yy = vertical distance

Horizontal Translation: (x+n,y)(x + n, y) for moving nn units right

Vertical Translation: (x,y+n)(x, y + n) for moving nn units up

Origin: (0,0)(0, 0)

💡Examples

Problem 1:

Identify the coordinates of a point that is 55 units to the right of the origin and 22 units above the origin.

Solution:

Step 1: Start at the origin (0,0)(0, 0). Step 2: Move 55 units to the right along the x-axis to reach x=5x = 5. Step 3: Move 22 units up parallel to the y-axis to reach y=2y = 2. Step 4: Combine the numbers into an ordered pair: (5,2)(5, 2).

Explanation:

The horizontal distance is the first number in the pair, and the vertical distance is the second number.

Problem 2:

A square has a vertex at (1,1)(1, 1). If you move this vertex 33 units to the right and 44 units up, what are the new coordinates?

Solution:

Step 1: Identify the initial xx and yy values: x=1,y=1x = 1, y = 1. Step 2: Add the horizontal movement to xx: 1+3=41 + 3 = 4. Step 3: Add the vertical movement to yy: 1+4=51 + 4 = 5. Step 4: The new coordinate is (4,5)(4, 5).

Explanation:

Translating a point involves adding or subtracting from the original coordinates based on the direction of movement.

Problem 3:

A triangle has vertices at A(1,1)A(1, 1), B(4,1)B(4, 1), and C(1,3)C(1, 3). If the entire triangle is translated 22 units to the right and 11 unit up, what are the new coordinates of vertex BB?

A triangle showing a translation from the original position to a new position B' at (6,2).

Solution:

The new coordinates of vertex BB are (6,2)(6, 2).

Explanation:

To find the new position, take the original coordinates of B(4,1)B(4, 1). Add 22 to the xx-coordinate (4+2=64 + 2 = 6) and add 11 to the yy-coordinate (1+1=21 + 1 = 2).

Problem 4:

Find the area of a rectangle whose corners are at (1,1)(1, 1), (5,1)(5, 1), (5,4)(5, 4), and (1,4)(1, 4) by calculating the length and width from the coordinates.

A rectangle on a grid with corners at (1,1), (5,1), (5,4), and (1,4).

Solution:

Area=12 square unitsArea = 12 \text{ square units}

Explanation:

Length = difference in xx-coordinates: 5−1=45 - 1 = 4 units. Width = difference in yy-coordinates: 4−1=34 - 1 = 3 units. Area=length×width=4×3=12Area = length \times width = 4 \times 3 = 12.