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Use coordinates to solve geometric problems

Coordinates represent a point’s horizontal and vertical positions relative to the origin, with the ordered pair (x,y)(x,y) distinguishing the xx-coordinate from the yy-coordinate across all four quadrants and the axes. These relationships support solving geometric problems involving plotted points, horizontal or vertical distances found by absolute differences, and the perimeter or area of simple rectangles and polygons; reversing coordinate order or ignoring negative signs changes the location. This understanding does not include slope, the distance formula for diagonal segments, or midpoint formulas.

Detailed Explanation: Use coordinates to solve geometric problems

An ordered pair (x,y)(x,y) tells a point’s location:

  • xx tells how far left or right the point is from the origin.
  • yy tells how far up or down the point is.
  • Always read the xx-coordinate first and the yy-coordinate second.
  • Keep negative signs. They show that a point is left of or below the origin.

Example: A rectangle has vertices

A(−3,1),B(2,1),C(2,−2),D(−3,−2).A(-3,1),\quad B(2,1),\quad C(2,-2),\quad D(-3,-2).

Find its perimeter and area.

Step 1: Find the horizontal length.

Points AA and BB have the same yy-coordinate, so the segment between them is horizontal. Use the absolute difference of the xx-coordinates:

∣2−(−3)∣=∣5∣=5. \vert 2-(-3) \vert = \vert 5 \vert =5.

The rectangle’s length is 55 units.

Step 2: Find the vertical width.

Points BB and CC have the same xx-coordinate, so the segment between them is vertical. Use the absolute difference of the yy-coordinates:

∣−2−1∣=∣−3∣=3. \vert {-2}-1 \vert = \vert -3 \vert =3.

The rectangle’s width is 33 units.

Step 3: Find the perimeter.

A rectangle has two lengths and two widths:

P=2(5)+2(3)=10+6=16.P=2(5)+2(3)=10+6=16.

The perimeter is 16 units\boxed{16\text{ units}}.

Step 4: Find the area.

Multiply the length by the width:

A=5⋅3=15.A=5\cdot 3=15.

The area is 15 square units\boxed{15\text{ square units}}.

Using absolute differences makes sure the distances are positive, even when coordinates are negative.

Learn by doing: Use coordinates to solve geometric problems

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