The squarest crossing of all — and the shortest way from a point to a line.
Of all the ways two lines can cross, one is special: the square crossing, where the four angles are each exactly 90°. That single idea — perpendicular — is everywhere once you look for it. It is the upright against the level, the height of a shape measured honestly, the corner of every door and book and tile. And it answers a question a plain segment cannot: how far is this point from that line? The shortest road there always runs straight across, at a right angle — and that road has a name we will earn in this lesson.
In 14.1 you saw that two crossing lines make a tied-together family of four angles: opposite angles are equal (vertical angles), and angles side by side on a straight line add to 180° (a linear pair). Most crossings are lopsided — a wide angle facing a wide one, a narrow facing a narrow. But suppose you tilt one line until just one of the four angles measures exactly 90°.
Watch what the rules from 14.1 do. If one angle is 90°, the angle next to it on the straight line is its linear-pair partner, so it is 180° − 90° = 90° as well. The angle across from the first is its vertical partner, so it is 90° too — and the last corner, also a vertical partner, is 90° once more. One right angle forces all four.
When two lines cross this way we call them perpendicular, and we write it with the upright symbol ⊥:
AB ⊥ CD
read aloud as "A B is perpendicular to C D." The point where they meet is called the foot of the perpendicular. On a figure we never write "90°" four times — we draw one little square in a corner, and that single mark tells the reader the whole crossing is square.
Two lines are perpendicular (⊥) when they cross at a 90° angle. You only have to check one of the four angles — the rest are pinned at 90° automatically. The meeting point is the foot of the perpendicular.
Pick any line, then pick any point — it can sit right on the line or float off it. Here is a quietly powerful fact:
Through that point there is exactly one line perpendicular to the given line.
Exactly one means two things at once: at least one such perpendicular always exists, and you can never find a second different one. You can feel this with a set-square: slide the square's right-angle corner along a ruler until its upright edge passes through your point. There is one resting place that works, and only one. Slide a hair past it and the edge misses the point.
The picture looks a little different in the two cases. When the point is on the line, the perpendicular stands straight up out of it like a flagpole. When the point is off the line, the perpendicular reaches down and touches the line at a single spot — the foot — and that act of drawing it is called dropping a perpendicular.
"How many perpendiculars can I draw through this point?" The answer is always the same: one. Not zero, not two — exactly one, whether the point sits on the line or away from it.
Now stand a point P above a line l and ask the practical question: of all the segments you could draw from P down to the line, which is the shortest? Try a slanted one over to the left, another over to the right — each is a little ramp down to the line. Slide the bottom end along and the ramp's length keeps changing.
There is a clear winner. The shortest segment is the one that meets the line at a right angle — the perpendicular segment PF, landing at the foot F straight below P. Every slanted segment PQ is the hypotenuse of a right triangle whose legs are PF and the bit of line from F to Q; the hypotenuse is always longer than the leg PF. The straight-across crossing beats every slanted one.
From a point off a line, the perpendicular segment to the line is shorter than every slanted segment. Its length is the smallest you can possibly get — and that minimum is exactly what we will call the distance.
Earlier you found the distance between two points: just the length of the plain segment joining them. But "distance from a point to a line" needs care, because a line offers infinitely many segments to choose from. We settle it with the winner we just found:
The distance from a point to a line is the length of the perpendicular segment from the point to the line.
We pick the perpendicular for a good reason — it is the shortest, so it is the only fair, single answer to "how far?" Any slanted segment would overstate the gap. And notice a tidy edge case: if the point happens to lie on the line, the foot is the point itself, the perpendicular segment shrinks to nothing, and the distance is 0.
Point → point: the length of the segment joining them.
Point → line: the length of the perpendicular segment — measured straight across, at 90°.
This is exactly how the real measurement is taken. A softball player standing on the baseline is "so many feet from the fence" — and you measure it straight out at a right angle to the fence, not along some slanting diagonal. A diver's height above the water, the clearance under a bridge, the width of a road: each is a perpendicular distance, because that is the honest, shortest gap.
Lines AB and CD cross, and one of the four angles measures 90°. What are the other three angles, and what can you write about the lines?
All three of the other angles are 90° (linear pair gives 180° − 90° = 90°, and the two vertical partners copy across). So AB ⊥ CD.
How many lines can you draw through a single point that are perpendicular to a given line?
Exactly one — whether the point is on the line or off it.
From a point off a line, you draw the perpendicular segment and also a slanted segment to the line. Which one is shorter?
The perpendicular segment is shorter. Every slant is the hypotenuse of a right triangle whose leg is the perpendicular, and a hypotenuse is always longer than a leg.
Fill in the blank: "The distance from a point to a line is the length of the ______ from the point to the line."
the perpendicular segment (the one that meets the line at a right angle).
From point P, the perpendicular segment to line l is 5 units long, and a slanted segment from P to l is 7 units long. What is the distance from P to l?
5 units. Distance to a line is always the perpendicular length — the 7-unit slant is longer and does not count.
True or false? The distance from a point that lies on a line to that line is 0.
True. The foot of the perpendicular is the point itself, so the perpendicular segment has length 0.
Six questions to lock it in. Tap the answer you think is right.
This lesson turns the angle family from 14.1 into a single, powerful special case — the right-angle crossing — and then uses it to define a genuinely new measurement: the distance from a point to a line. The throughline is "one right angle forces all four," followed by the existence-and-uniqueness of the perpendicular, and finally the minimizing property that justifies the definition of distance.
The misconception to watch is measuring point-to-line distance along a slant instead of the perpendicular. Students often pick whatever segment is drawn or whichever one "looks reasonable." Reinforce that distance to a line is, by definition, the shortest segment — and the shortest is always the perpendicular. The shortest widget is built to make that minimum visible: the slant only equals the distance when it lands exactly on the foot.
Common Core alignment: 4.G.A.1 (draw and identify perpendicular lines), 7.G (geometric figures and angle reasoning), and the high-school standards G-CO.A.1 (precise definition of perpendicular lines) and G-CO.D.12 (construct the perpendicular to a line through a point). The uniqueness of the perpendicular and the perpendicular-as-shortest property are the foundation for later work on distance, altitude, and the distance from a point to a line in coordinate geometry.