ACT Math: Apply Distance and Midpoint Formulas to Coordinate Geometry Problems

Distance formula: d=sqrt((x2-x1)^2+(y2-y1)^2). Midpoint formula: M=((x1+x2)/2,(y1+y2)/2). Example: Find the distance between (1,2) and (4,6). d=sqrt((4-1)^2+(6-2)^2)=sqrt(9+16)=sqrt(25)=5. Example: Find the midpoint of the same two points. M=((1+4)/2,(2+6)/2)=(2.5, 4). These two formulas unlock most coordinate geometry problems on ACT Math; memorizing them and practicing them until automatic saves enormous time.

Example: A circle has center (2,3) and passes through (5,7). Find the radius. Radius=distance from center to point=sqrt((5-2)^2+(7-3)^2)=sqrt(9+16)=5. The distance formula immediately gives the radius without additional steps.

Trap 1: Forgetting to square the differences before adding. d=sqrt((x2-x1)^2+(y2-y1)^2), not sqrt(x2-x1+y2-y1). Forgetting the squares gives wrong distances. Trap 2: Forgetting to take the square root at the end. Some students calculate the sum of squares but forget to sqrt it. Write out the formula explicitly every time to prevent this careless error. When you use the distance formula, write it out: d=sqrt((...squared)+(...squared)). Then calculate step by step. This explicit work prevents errors.

Before you lock in a distance answer, verify: Did I square the differences? Did I add them? Did I take the square root? All three steps are essential.

Problem 1: Find the distance between (-1,3) and (2,7). d=sqrt((2-(-1))^2+(7-3)^2)=sqrt(3^2+4^2)=sqrt(9+16)=5. Problem 2: Find the midpoint of (0,0) and (8,4). M=((0+8)/2,(0+4)/2)=(4,2). Problem 3: A segment has endpoints (1,2) and (9,8). Find its length and midpoint. Length: d=sqrt((9-1)^2+(8-2)^2)=sqrt(64+36)=sqrt(100)=10. Midpoint: M=((1+9)/2,(2+8)/2)=(5,5). Problem 4: Two circles: center A at (0,0) radius 5, center B at (3,4) radius r. If the circles are externally tangent (touching at one point), what is r? Distance between centers=5+r (sum of radii). Distance AB=sqrt((3-0)^2+(4-0)^2)=sqrt(9+16)=5. So 5=5+r, r=0. This means the circles don't work as described; reconsider the problem setup. All four use distance and midpoint formulas in different contexts.

Do ten more coordinate geometry problems daily. By test day, these formulas will be automatic and you'll apply them correctly every time.

Distance and midpoint formulas appear in multiple ACT Math topics (geometry, coordinate plane, occasionally word problems). Once you master these formulas and apply them automatically, you'll solve coordinate geometry problems quickly and accurately, earning reliable points.

Memorize these two formulas this week. Drill them daily until they're automatic. By test day, using them will be second nature and you'll solve coordinate problems in seconds.