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Not closer to one, not closer to the other—exactly in the middle.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"In coordinate geometry, that \"middle\" has specific x and y values. The midpoint's x-coordinate lands exactly between the two x-values you started with. Same for y. That's really all there is to it.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Here's why this matters beyond textbook problems: midpoints show up whenever you need to find a center. Centering a picture frame between two hooks. Finding where two diagonal lines cross in a rectangle. Placing a label at the middle of a line segment in a diagram. The concept is everywhere once you start noticing it.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The Midpoint Formula","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The formula to find the midpoint M between two points (x₁, y₁) and (x₂, y₂) is:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"That's it. Add the x-values, divide by 2. Add the y-values, divide by 2. You're averaging.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Why does averaging work? Think about it: the average of 10 and 20 is 15—right in the middle. The average of 0 and 100 is 50—again, dead center. Averaging always lands you exactly between two values. The midpoint formula just applies that logic to both coordinates at once.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Pro tip:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" This formula is worth memorizing. It's short, it makes sense, and it appears on nearly every standardized math test. You'll use it dozens of times before you're done with geometry.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"How to Use This Calculator","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Step 1:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Enter the coordinates of your first point. That's your x₁ value in the first box, y₁ in the second.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Step 2:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Enter the coordinates of your second point—x₂ and y₂.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Step 3:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Your midpoint appears instantly.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The calculator works with any numbers: positive, negative, whole numbers, decimals, even fractions expressed as decimals. Plug in your values and let it handle the arithmetic so you can focus on understanding the problem.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Worked Examples","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Let's walk through several scenarios so you can see the pattern.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Example 1: Basic Positive Coordinates","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Problem:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Find the midpoint between (2, 4) and (8, 10).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Solution:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint x = (2 + 8) / 2 = 10 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"5","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint y = (4 + 10) / 2 = 14 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"7","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Answer:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The midpoint is ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"(5, 7)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Notice how 5 sits exactly between 2 and 8, and 7 sits exactly between 4 and 10. The formula delivers exactly what you'd expect.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Example 2: Negative Coordinates","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Problem:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Find the midpoint between (-6, 3) and (4, -1).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This is where students often slip up. Negatives aren't hard—just stay careful with your signs.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Solution:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint x = (-6 + 4) / 2 = -2 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"-1","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint y = (3 + (-1)) / 2 = 2 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"1","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Answer:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The midpoint is ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"(-1, 1)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The midpoint landed in a different quadrant than either original point. That's completely normal when your points span across axes.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Example 3: Decimal Values","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Problem:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Find the midpoint between (1.5, 2.8) and (4.5, 6.2).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Solution:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint x = (1.5 + 4.5) / 2 = 6 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"3","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint y = (2.8 + 6.2) / 2 = 9 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"4.5","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Answer:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The midpoint is ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"(3, 4.5)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Decimals follow the same process. Your midpoint might come out as a whole number, a decimal, or a fraction—all valid.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Example 4: Vertical or Horizontal Line Segments","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Problem:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Find the midpoint between (0, 5) and (0, 11).","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Solution:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint x = (0 + 0) / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"0","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint y = (5 + 11) / 2 = 16 / 2 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"8","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Answer:","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The midpoint is ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"(0, 8)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Both points have x = 0, so they form a vertical line along the y-axis. The midpoint stays on that same vertical line. You'll see this pattern whenever your original points share an x-value or a y-value.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Where Midpoints Show Up in Real Life","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Hanging art on a wall.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" You've got two hooks 48 inches apart and want to center a picture between them. The midpoint tells you to place the center of the frame at 24 inches from either hook. Same math, practical result.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Designing graphics or websites.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Need to place a button exactly between two elements? Developers calculate midpoints constantly when positioning objects in layouts, games, or animations.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Meeting someone halfway.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" If you're at coordinates (2, 8) on a city grid and your friend is at (10, 2), the midpoint (6, 5) is the fair meeting spot—assuming straight-line travel.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Construction layout.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" A contractor centering a door in a wall opening, a carpenter finding the middle of a board for a cut, an architect placing a column between two reference points—all midpoint problems in disguise.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Finding the center of a line segment for geometry proofs.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Many proofs require you to establish a midpoint before proving triangles congruent or constructing perpendicular bisectors. The midpoint is often your first step.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Common Mistakes (and How to Avoid Them)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Forgetting to divide by 2.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" This is the most common error. You add your coordinates correctly, then forget the second half of the formula. Always divide by 2—you're averaging, not just adding.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Mixing up x and y values.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Keep your coordinates organized. The first number in a pair is always x (horizontal), the second is y (vertical). Write them down separately if it helps.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Sign errors with negatives.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Adding -6 and 4 gives you -2, not 10. If negatives trip you up, slow down and write out each step. One wrong sign throws off your entire answer.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Assuming the midpoint is \"between\" the coordinates visually.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The midpoint might have a larger x-value than both original points if you're working with negatives. Trust the math, not your intuition about what \"between\" looks like on a number line.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"What About 3D Coordinates?","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The formula extends naturally to three dimensions. If you're working with points that have x, y, and z coordinates:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Same idea—average each coordinate separately. You'll encounter this in 3D geometry, physics problems, computer graphics, and CAD applications. 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They often appear in the same problem set.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Slope.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The slope between your two original points stays the same when measured from either point to the midpoint. The midpoint lies on the line—it doesn't change the line's direction.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Perpendicular Bisector.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" A line that passes through the midpoint at a 90-degree angle to the original segment. Finding the midpoint is step one of constructing a perpendicular bisector.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A Note on Accuracy","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"This calculator uses the standard Euclidean midpoint formula for flat coordinate planes. The results are mathematically exact.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For geographic coordinates over long distances (like finding the midpoint between two cities), Earth's curvature matters, and you'd need spherical geometry instead. 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34:["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-8","children":[["$","h2",null,{"className":"typo-large mb-6 text-mist-950","children":"Frequently Asked Questions"}],["$","div",null,{"className":"space-y-6","children":[["$","div","0",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What is the midpoint formula?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The midpoint formula is M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2). You're averaging: add the x-coordinates and divide by 2, then add the y-coordinates and divide by 2. The result is the exact center point between your two coordinates.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","1",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How do I find the midpoint when one or both coordinates are negative?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Use the same formula—negatives don't change the process. If x₁ = -4 and x₂ = 2, then (-4 + 2) / 2 = -2 / 2 = -1. Just watch your signs when adding. Writing out each step helps prevent errors.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","2",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can I find an endpoint if I only know the midpoint and one endpoint?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Yes. Rearrange the formula: if M is the midpoint and A is the known endpoint, then the other endpoint B = (2·Mₓ - Aₓ, 2·Mᵧ - Aᵧ). You're essentially \"undoing\" the average.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","3",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What's the difference between midpoint and distance?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint gives you a location (two coordinates). Distance gives you a length (one number). The midpoint is ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"where","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" the center is; the distance is ","type":"text","version":1},{"detail":0,"format":2,"mode":"normal","style":"","text":"how far","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" the points are from each other.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","4",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Does this formula work in three dimensions?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Absolutely. For 3D points, add a z-term: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2). Same logic, one more coordinate.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","5",{"className":"pb-6 last:pb-0","children":["$L36","$L37"]}],"$L38","$L39","$L3a","$L3b"]}]]}]
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28:null
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36:["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Why does averaging give you the midpoint?"}]
37:["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The average of two numbers always lands exactly between them—that's what an average does. Since coordinates describe positions, averaging the x-values finds the horizontal center, and averaging the y-values finds the vertical center. Together, you get the center point.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]
38:["$","div","6",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Is the midpoint the same as a centroid?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Not quite. A midpoint is the center of a line segment (two points). A centroid is the center of mass of a polygon—for a triangle, you'd average all three vertices, not just two. Related concepts, different applications.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
39:["$","div","7",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What if both points are the same?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Then the midpoint equals that point. If you start at (3, 5) and end at (3, 5), you haven't moved—so the \"middle\" is just (3, 5). The formula still works: (3+3)/2 = 3, (5+5)/2 = 5.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3a:["$","div","8",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can the midpoint formula give wrong answers?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The formula is mathematically exact—no rounding, no estimation. If your answer seems wrong, double-check your arithmetic, especially with negative signs. The formula itself doesn't fail.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3b:["$","div","9",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How is this used on standardized tests?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Midpoint questions appear on the SAT, ACT, GRE, and most state geometry exams. Sometimes you'll calculate a midpoint directly; other times you'll use it as an intermediate step to find distances, slopes, or missing vertices of shapes.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
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