ACT Math: Midpoint and Distance Formulas Made Simple

Midpoint formula finds the center point between two coordinates: M=((x1+x2)/2, (y1+y2)/2). Use this when the question asks for "midpoint," "center," or "halfway point." Distance formula finds how far apart two points are: d=sqrt((x2-x1)^2+(y2-y1)^2). Use this when the question asks for "distance," "length," or "how far." The confusion arises because both formulas use the same two points, but they answer different questions. Before you calculate, read the question and underline whether it asks for a point (midpoint) or a number (distance).

Example: Points A(2,3) and B(8,9). Midpoint: ((2+8)/2, (3+9)/2)=(5,6). Distance: sqrt((8-2)^2+(9-3)^2)=sqrt(36+36)=sqrt(72)=6*sqrt(2). The midpoint is a coordinate pair; the distance is a number. If you reverse them, you lose the point.

Error 1: Forgetting to divide by 2 in the midpoint formula. Fix: Highlight the division sign in your formula and check that you divided both coordinates. Error 2: Forgetting to square the differences in the distance formula. Fix: Write out (x2-x1)^2 explicitly; don't skip the squaring step. Error 3: Forgetting to take the square root at the end of the distance formula. Fix: Your final distance answer should always be under a radical or a decimal; if it's a perfect square (like 36), take the square root (6). Write the formula out in full every single time; shortcutting leads to these three errors.

Self-check routine: After you calculate midpoint, verify that your answer is actually between the two original points. After you calculate distance, verify that your answer is positive and reasonable for the coordinates given.

Problem 1: Find the midpoint of (1,4) and (5,10). Problem 2: Find the distance between (0,0) and (3,4). Problem 3: Find the midpoint of (-2,6) and (4,2). Problem 4: Find the distance between (1,1) and (4,5). For each, write the full formula, plug in the numbers, and show every step. For Problems 1 and 3, your answer should be an ordered pair. For Problems 2 and 4, your answer should be a distance (possibly with a radical). Do all four by hand without a calculator, then verify using a calculator to check your arithmetic.

Answers: P1: (3,7). P2: 5. P3: (1,4). P4: 5. If you missed any, redo that problem step-by-step and identify where the arithmetic or formula application broke down.

Midpoint and distance formula questions appear on nearly every ACT Math test (questions 25-45). Unlike harder geometry problems, these questions are purely mechanical: plug in numbers, follow the formula, and calculate. Once you memorize the two formulas and avoid the three errors, you'll answer these questions faster and more accurately than most test-takers.

This week, memorize the two formulas and drill one problem per day. By test day, these formulas will be so automatic that you'll solve midpoint and distance problems in under 1 minute each, gaining significant time for harder questions.