Coordinate geometry drills

Plot points and find distance and midpoint on the coordinate plane.

Free, timed SHSAT practice sets on coordinate geometry— every question includes a full worked explanation, and there's no account required.

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How to solve coordinate geometry problems

Most coordinate questions use one of three tools. Example points: \((1, 2)\) and \((7, 10)\).

To find a missing endpoint from the midpoint, take the same step again: from \(A\) to \(M\), then the same distance from \(M\) to \(B\).

Common traps: forgetting to divide by 2 for a midpoint, adding the across and up distances (\(6 + 8 = 14\)) instead of using the square root, and forgetting the \(\dfrac{1}{2}\) in a triangle's area.

Worked examples

Example 1.
What is the distance between the points \((-6, -2)\) and \((-3, 2)\)?
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Answer:
A

Distance \(= \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\) units. (Stopping at \(25\) forgets the square root. Adding \(3 + 4\) counts the steps along the grid, 7, not the straight-line distance. Subtracting the squares instead of adding them gives \(\sqrt{7}\).)

Example 2.
\(M(-4, 3)\) is the midpoint of segment \(AB\), and \(A = (6, -1)\). What are the coordinates of \(B\)?
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Answer:
C

Since \(M\) is the midpoint, \(B = (2m_x - x_1, 2m_y - y_1) = (2(-4) - 6, 2(3) - (-1)) = (-14, 7)\). (Averaging \(A\) and \(M\) gives the midpoint of \(AM\), (1, 1). Subtracting \(M - A\) gives only the step from \(A\) to \(M\), (-10, 4). Doubling \(A\) and subtracting \(M\) mixes up their roles, giving (16, -5).)

Example 3.
A triangle has vertices \((0, 0)\), \((4, 6)\), and \((2, 7)\). What is the area of the triangle, in square units?
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Answer:
8

Draw the \(4 \times 7\) rectangle with corners \((0, 0)\) and \((4, 7)\): area 28. Subtract the three right triangles around the shape: \(\dfrac{1}{2}(4)(6) + \dfrac{1}{2}(2)(7) + \dfrac{1}{2}(2)(1) = 20\). Area \(= 28 - 20 = 8\) square units.