Left: lines AB and CD cross squarely, so AB ⊥ CD — one right-angle mark, and the other three corners are right angles too. Right: from point P the green segment straight down to the foot F is the shortest path — it is the distance from P to the 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.
14.2.1 Perpendicular lines
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.
Mark one corner with a right-angle square and the other three follow for free: the linear-pair partner is 180° − 90°, and each vertical partner copies its opposite. So AB ⊥ CD at the foot O.
Key idea
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.
14.2.2 The perpendicular through a point — it exists, and it's the only one
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.
Try it The one and only perpendicular
Switch between a point on the line and a point off it. Either way, exactly one perpendicular fits — the red dashed slant is an impostor.
Point P is
In plain words
"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.
14.2.3 The perpendicular segment is the shortest
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 segmentPF, 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.
Try it Straight across wins
Drag the point Q along the line and watch the slant PQ change length. It bottoms out exactly when Q reaches the foot F — that shortest value is the perpendicular PF.
Slide Q
Key idea
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.
14.2.4 Distance from a point to a line
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.
Two kinds of distance
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.
The runner on the baseline stands a perpendicular distance from the fence — measured straight across at 90° (the green segment), never along a slant.
★ Recap
Perpendicular (⊥): two lines that cross at 90°. Checking one right angle is enough — all four become 90° by the vertical-angle and linear-pair rules.
The crossing point is the foot of the perpendicular; mark a square, not "90°", in the corner.
Through any point — on or off a line — there is exactly one perpendicular to that line.
The perpendicular segment from a point to a line is the shortest of all segments to that line.
Distance from a point to a line = the length of that perpendicular segment. (If the point is on the line, the distance is 0.)
✎ Exercises
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?
Answer
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?
Answer
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?
Answer
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."
Answer
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?
Answer
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.
Answer
True. The foot of the perpendicular is the point itself, so the perpendicular segment has length 0.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.
eastmath.com · Stage 14 · 14.2 Perpendicularity & Distance · Reasoning, one step at a time
eastmath.com · 14.2 Perpendicularity and Distance · 14.2.3 The perpendicular segment is the shortest