ACT® · Math: Integrating Essential Skills
The Coordinate Plane
The coordinate plane turns geometry into arithmetic: midpoint averages coordinates, distance builds a right triangle from the legs, and plotting vertices lets you find area directly.
The coordinate plane gives every point an address, (x, y), and that address lets you do geometry with arithmetic instead of a protractor. The midpoint formula, ((x1+x2)/2, (y1+y2)/2), is just an average of the two x's and the two y's, because the point exactly between two others splits the difference in each direction. The distance formula, √((x2-x1)2 + (y2-y1)2), is the Pythagorean theorem in disguise: the horizontal gap and vertical gap between two points are the legs of a right triangle, and the distance between the points is the hypotenuse.
This same leg-based thinking lets you find area without memorizing a shape formula. If a triangle has a horizontal leg and a vertical leg meeting at a right angle, those legs ARE the base and height, so area = (1/2) x base x height works directly off the coordinates. For a right triangle running from (0,0) to (6,0) to (6,8), the base is 6 and the height is 8, giving area 24. Rectangles and other polygons on the grid work the same way: read the side lengths straight off the differences in x and y.
Reflections connect this to transformations elsewhere in geometry. Reflecting a point across the line y = x swaps its coordinates, turning (a, b) into (b, a), because that mirror line sits at a 45-degree angle and flips horizontal and vertical roles. Keep these tools together: distance and midpoint describe segments, leg-based area describes regions, and reflection shows how coordinates move under symmetry.
Key terms
- Midpoint
- The point exactly halfway between two points, found by averaging their x-coordinates and their y-coordinates.
- Distance formula
- √((x2-x1)2+(y2-y1)2), the Pythagorean theorem applied to the horizontal and vertical gaps between two points.
- Leg
- One of the two sides forming the right angle in a right triangle, usable directly as base or height.
- Reflection across y = x
- A flip over that diagonal line that swaps a point's x and y coordinates.
Watch it explained
- How it worksHow does plotting a polygon's vertices let you find its area with no area formula at all?
- Worked exampleFind the distance between the points (-3, 4) and (5, -2) in the coordinate plane.
- Worked exampleFind the area of a right triangle in the coordinate plane with legs running from (0, 0) to (6, 0) and then up to (6, 8).
- Going furtherWhat happens to a point's coordinates when its shape is reflected across the line y = x?
Check yourself
End of topic.