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I have a web application where I need some functionality to draw two points that is the start of a line, and then extend that line to a given length (that is given by a number in meters). I know how to create the line as a polyline based on the 2 points, but how can I extend this by the given distance?

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Just calculate the slope of your 2 points, then use geometry to calculate the "new" second point given the slope and the first point and the new distance (d).

  // returns a graphic with a polyLine geometry
  function setDistance(point, slope, d) {
        var symbol = new SimpleLineSymbol();

    // https://math.stackexchange.com/a/656512/42151
    var r = Math.sqrt(1 + Math.pow(slope, 2));
    var newX = point.x + (d/r);
    var newY = point.y + ((d * slope)/r);
    return new Graphic(new Polyline({paths:[[[point.x, point.y], [newX, newY]]], "spatialReference":{wkid: 102100}}), symbol);
  }

Here's an example

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