This lesson is about measuring distance in the most direct way possible: straight at a line, at a right angle. It has two parts. First you will find the distance from a single point to a line. Then you will use that same skill to find the distance between two parallel lines — and see why that distance is the same no matter where you measure it.
Distance from a point to a line: the length of the perpendicular segment from the point to the line. The point where the perpendicular segment intersects the line is called the intersection point. Of all the segments that can be drawn from the point to the line, the perpendicular segment is the shortest, which is why it is the one that counts as the distance.
Parallel lines are lines that lie in the same plane and do not intersect. The distance between two parallel lines is the distance from any point on one line to the other line. Because the lines never get closer together or farther apart, that distance is the same everywhere along the lines.
The distance formula: the distance between (x1, y1) and (x2, y2) is √[(x2 − x1)2 + (y2 − y1)2]. Distances on this page are rounded to one decimal place, the same style used in class.
The problem: Find the distance from the point (1, 0) to the line y = −x + 3.
What to do: Press Next Step to work through the method one stage at a time. Each step is drawn on the grid as it is named, so you can watch the figure build. Press Restart at any time to start the example over.
What to do: In this activity, the orange handle sits on the given line. Drag it along the line. The segment from the point to the handle changes length as you drag — watch the readout. The segment only earns the name distance when it is perpendicular to the line. When you find that position, the segment settles exactly onto the intersection point, a right-angle marker appears, and the activity confirms the distance. Press New Challenge for a new point and line.
What to do: The two parallel lines in this activity are joined by a perpendicular segment. Drag the orange handle to slide the segment along the lines. Notice the segment moves, both of its endpoints move, and yet its length never changes — that is the key idea of this half of the lesson. It is also why Step 2 of the method works: since the distance is the same everywhere, you may pick whichever point on the first line is easiest.
Choose an example, then press Show the Steps to see the full 5-step method carried out for it, with the same numbers the activity is using.
Work through these six questions, one at a time. They follow the method from this page: recognize which segment measures the distance, find a perpendicular slope, find the intersection point, and then carry the full method through to a distance. Diagrams are provided where they help. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.
Distance Between Two Parallel Lines — Mr. J Geometry. All activities on this page run entirely in your browser.