Chords, secants and tangents
A straight line and a circle lie on the same page. In how many points can they meet, and what decides which?
Where you are 15 in Circles
Why this exists
A circle already has one named segment that runs across it: a chord, with both ends on the curve. But a segment stops at its ends, and a straight line does not. Lay a ruler across a coin and three different things can happen. The ruler can cut straight across, going in at one point and out at another. It can rest against the edge and touch it at a single point. Or it can lie beside the coin and miss it altogether.
Those three cases have names, secant, tangent and a line that misses, and every later fact about circles is phrased in them: the angles that chords and tangents make, the lengths of their pieces, the right angle a tangent makes with a radius. This lesson sets out the words and answers the question the coin raises: before you draw anything, how can you tell which case you are in? One number decides it, the distance from the center to the line, compared with the radius.
Definition
Take a circle with center O and radius r, and a straight line in the same plane.
- A chord is a segment whose two endpoints both lie on the circle. A chord that passes through the center is a diameter.
- A secant is a line that meets the circle at exactly two points. The part of a secant between those two points is a chord, and every chord lies on exactly one secant, the line through its two ends. The difference is that a chord is a segment and has a length, while a secant is a line and runs on past the circle in both directions.
- A tangent is a line that meets the circle at exactly one point. That point is the point of tangency, and the line is said to be tangent to the circle at that point. A segment or ray is also called tangent when it lies along a tangent line and contains the point of tangency.
- A line can also meet the circle at no points. It has no special name; it misses.
- Concentric circles are circles in the same plane that share a center and have different radii.
One more measurement is needed. The distance from the center to a line is the length of the perpendicular segment from O to the line. Its foot, the point F where it meets the line, is the point of the line nearest to O, which is why distance to a line is always measured this way; see distance from a point to a line. In this lesson that distance is called d.
The line through two named points A and B is called line AB, the segment between them is written \overline{AB}, and the circle with center O is \odot O.
Intuition
Hold a ruler far below a coin and slide it slowly upward, keeping it level. At first it misses. At the instant its edge reaches the coin it touches at one point, the lowest point of the coin. Slide it further and it cuts across: it goes in at one point and out at another, and the piece of the edge lying over the coin is a chord. That chord grows as the ruler nears the center, is longest when the ruler passes through the center, where it is a diameter, and then shrinks again until the ruler touches the top of the coin and leaves it.
Through all of that, the only thing that changed was how far the ruler was from the center. While that distance was more than the radius, the ruler missed. When it equaled the radius, the ruler touched. When it was less, the ruler cut across. That is the whole lesson, and the results below make it exact.
Results
Result 1 (at most two points). A line and a circle in the same plane meet at two points, at one point, or at none. They never meet at three or more.
Result 2 (the distance test). Let d be the distance from the center of a circle of radius r to a line \ell. Then:
| Distance | Points in common | The line is |
|---|---|---|
| d < r | 2 | a secant |
| d = r | 1, the foot F | a tangent |
| d > r | 0 | a line that misses |
Result 3 (where the two points sit). When d < r, the two points lie on either side of the foot F, each at distance \displaystyle \sqrt{r^2 - d^2} from it. So F is the midpoint of the chord that the secant cuts off, and that chord has length
Two facts are hiding in these results, and each has its own lesson. In the tangent case the point of tangency is the foot F, so the radius to it is perpendicular to the tangent: that is where the tangent properties begin. And in the secant case the perpendicular from the center lands at the midpoint of the chord, which is where the chord theorems begin.
Why it is true
Let F be the foot of the perpendicular from O to the line \ell, so that OF = d. Take any point X on \ell other than F. The triangle OFX has a right angle at F, so by the Pythagorean theorem
The same equation also holds when X = F, where FX = 0, and when O lies on \ell itself, where d = 0 and OX = FX. So it holds for every point X of the line.
A point X of the line is on the circle exactly when OX = r, that is, when OX^2 = r^2. Put that into the equation and subtract d^2 from both sides:
Now look at the sign of the right-hand side.
- If d > r, then r^2 - d^2 is negative. No square is negative, so no point X works, and the line misses the circle.
- If d = r, then FX^2 = 0, so FX = 0 and X = F. Exactly one point works, the foot itself. The line is a tangent and F is the point of tangency.
- If d < r, then r^2 - d^2 is positive and \displaystyle FX = \sqrt{r^2 - d^2}. On a line there are exactly two points at a given positive distance from F, one on each side of it. So exactly two points work, the line is a secant, and F sits halfway between them.
That proves Results 2 and 3. Every case gives at most two points, which proves Result 1 as well.
The equation says one more thing: OX^2 = d^2 + FX^2 is never less than d^2, and equals it only at X = F. So the foot really is the nearest point of the line to the center, which is what made d the right distance to use.
Worked examples
Naming chords, secants and tangents
In Figure 4, K is the center of the circle and the points P, Q, R, S, T, U and V lie on it, while W lies outside it. Say what each of these is: \overline{PQ}, the line RS, the line \ell, \overline{KU} and \overline{VW}.
\overline{PQ} is a segment with both ends on the circle, so it is a chord. It does not pass through K, so it is not a diameter.
The line RS meets the circle at the two points R and S, so it is a secant. The part of it between R and S, the segment \overline{RS}, is a chord.
The line \ell meets the circle only at T, so it is a tangent, and T is the point of tangency.
\overline{KU} has one end at the center and one on the circle, so it is a radius. It is not a chord, because K is not on the circle.
\overline{VW} has only one end on the circle, so it is not a chord. It is not a secant or a tangent either, because those words name lines, and \overline{VW} is a segment that stops at V and W. The whole line through V and W does meet the circle, at V at least, so it is a secant or a tangent, but the segment itself is neither.
Counting points from the distance
A circle has radius 5 cm. Four lines lie at distances 3 cm, 5 cm, 7.5 cm and 0 cm from its center. How many points does each line share with the circle? For each line that crosses, how long is the chord it cuts off?
SolutionCompare each distance d with the radius r = 5 (Result 2).
- d = 3: since 3 < 5, the line is a secant and meets the circle at two points. By Result 3 the chord has length \displaystyle 2\sqrt{5^2 - 3^2} = 2\sqrt{16} = 8 cm.
- d = 5: the distance equals the radius, so the line is a tangent and meets the circle at one point.
- d = 7.5: since 7.5 > 5, the line misses, with no points in common.
- d = 0: the line passes through the center, and 0 < 5, so it is a secant. Its chord is \displaystyle 2\sqrt{5^2 - 0^2} = 10 cm, which is the diameter, as it must be for a chord through the center.
Lines on a grid
A circle has center C(1, 2) and radius 5. For each of the lines y = 5, x = 6 and y = -4, find its distance from C, decide whether it is a secant, a tangent or neither, and find the points it shares with the circle.
The line y = 5 is horizontal, so the perpendicular from C to it is vertical and lands at F(1, 5). The distance is the vertical gap, d = 5 - 2 = 3. Since 3 < 5, the line is a secant. By Result 3 the two points are \displaystyle \sqrt{5^2 - 3^2} = 4 to the left and right of F, at (-3, 5) and (5, 5).
Check. From C(1, 2) to (5, 5) is 4 across and 3 up, and \displaystyle \sqrt{4^2 + 3^2} = 5, so the point is on the circle.
The line x = 6 is vertical, so the perpendicular from C is horizontal and lands at (6, 2). The distance is 6 - 1 = 5, equal to the radius, so the line is a tangent, and the point of tangency is the foot, (6, 2).
The line y = -4 is horizontal, and its distance from C is 2 - (-4) = 6. Since 6 > 5, the line misses the circle. In the language of the proof, a point on it would need FX^2 = 5^2 - 6^2 = -11, and no square is negative.
Two concentric circles and one line
Two concentric circles have center O and radii 3 and 7. A line lies at distance d from O. For d = 2, 3, 5, 7 and 8, how many points does the line share with the two circles together?
The two circles share a center, so a line has the same distance d from both of them. Apply the distance test to each circle separately and add.
| d | Inner circle, r = 3 | Outer circle, r = 7 | Total |
|---|---|---|---|
| 2 | 2 < 3: secant, 2 | 2 < 7: secant, 2 | 4 |
| 3 | 3 = 3: tangent, 1 | 3 < 7: secant, 2 | 3 |
| 5 | 5 > 3: misses, 0 | 5 < 7: secant, 2 | 2 |
| 7 | 7 > 3: misses, 0 | 7 = 7: tangent, 1 | 1 |
| 8 | 8 > 3: misses, 0 | 8 > 7: misses, 0 | 0 |
The lines \ell_1 to \ell_5 in Figure 6 are these five cases, in order. Notice that no row has two tangents. A line tangent to both circles would need d = 3 and d = 7 at once, and one line has only one distance from O.
Which horizontal lines cross
A circle has center (2, -1) and radius 4. For which values of k is the line y = k a secant, for which is it a tangent, and for which does it miss?
SolutionThe line y = k is horizontal, so its distance from the center is the vertical gap between the heights k and -1:
It is a secant when d < 4:
It is a tangent when |k + 1| = 4, so k + 1 = 4 or k + 1 = -4, giving k = 3 or k = -5. The points of tangency are the feet of the perpendiculars from the center, straight above and below it: (2, 3) and (2, -5).
It misses when |k + 1| > 4, that is, when k < -5 or k > 3.
Check. The circle reaches 4 above and 4 below its center, so its highest point has height -1 + 4 = 3 and its lowest -1 - 4 = -5. Level lines between those heights cross it, level lines at those heights touch it, and the rest miss.
Lines through a point inside
A circle has center O and radius 5, and P is a point inside it with OP = 3. (a) Show that every line through P is a secant. (b) Of all the chords through P, which is the longest and which the shortest, and how long are they?
(a) Take any line \ell through P, and let F be the foot of the perpendicular from O to it. If F = P, the distance from O to \ell is OP = 3. If F \ne P, the triangle OFP has a right angle at F, so \overline{OP} is its hypotenuse, the longest side, and OF < OP = 3. Either way d \le 3 < 5, and by Result 2 the line is a secant. So no line through a point inside a circle can be a tangent or miss.
(b) By Result 3, the chord on a line at distance d has length \displaystyle 2\sqrt{25 - d^2}. As d grows, 25 - d^2 shrinks, so the chord gets shorter. From part (a), d can be anything from 0 up to 3.
- The smallest distance, d = 0, comes from the line through O. Its chord is the diameter, of length \displaystyle 2\sqrt{25} = 10. That is the longest chord through P.
- The largest distance, d = 3, comes from the line whose foot is P itself, the line through P at right angles to \overline{OP}. Its chord has length \displaystyle 2\sqrt{25 - 9} = 2\sqrt{16} = 8. That is the shortest chord through P, chord \overline{AB} in Figure 7.
So every chord through P is between 8 and 10 long.
A ring cut by a tangent chord
Two concentric circles have center O and radii 5 cm and 13 cm. A chord \overline{AB} of the larger circle is tangent to the smaller circle at M. Find the length of \overline{AB}, and use it to find the area of the ring between the two circles.
The line AB is tangent to the smaller circle, so by Result 2 its distance from O equals that circle’s radius, d = 5, and the point of tangency M is the foot of the perpendicular from O. For the larger circle the same line has d = 5 < 13, so it is a secant, and by Result 3 the half-chord is
So AB = 2 \times 12 = 24 cm.
Now the ring. Using the area of a circle, its area is the large disc minus the small one:
The bracket 13^2 - 5^2 is exactly the MB^2 found above. The same thing happens for any two concentric circles with radii R > r: the half-chord satisfies MB^2 = R^2 - r^2, so
The area of the ring depends only on the length of that one chord. Two rings with very different radii have the same area if their tangent chords are equally long.
Boundaries
The line and the circle lie in one plane. In space, a line can pass through a single point of a circle by piercing the plane the circle lies in. It meets the circle exactly once but is not a tangent. Everything here assumes one plane.
“Exactly one point” defines a tangent for circles only. For other curves it fails. The vertical line x = 0 meets the parabola y = x^2 at one point and crosses straight through it, and a line can be tangent to a wavy curve at one place and cut it somewhere else. Tangency to a general curve is defined differently, through the direction of the curve at the point.
A chord is a segment and a secant is a line. A chord has a length; a secant does not. A segment with only one end on the circle is neither, and a radius is not a chord, because the center is not on the circle.
The distance is the perpendicular distance. The distance from the center to some point of the line says nothing on its own: every line has points far from the center. Only the perpendicular distance d, to the nearest point, decides the case.
Concentric means same plane, same center, different radii. Two circles with the same center and the same radius are the same circle.
The radius is positive. A “circle” of radius 0 is a single point, and the three cases collapse.
Where this goes next
With the words in place, the circle theorems can be stated. The tangent case, where the point of tangency is the foot of the perpendicular, becomes the tangent properties: a tangent is perpendicular to the radius at the point of tangency, and the two tangents from an outside point are equally long. The secant case, where the foot is the midpoint of the chord, becomes the chord theorems, which go on to show that chords equally far from the center are equally long.
A chord also splits the circle into two arcs, and the region between a chord and its arc is the segment measured in sectors and segments. When two chords cross, or two secants meet outside the circle, the pieces they cut have a neat product rule, segment lengths in circles, and the angles they make are measured by the arcs they cut off, in angles made by chords, secants and tangents.
On a coordinate grid, the distance test has an algebraic twin. Putting a line into the equation of a circle gives a quadratic equation, and its two, one or no real solutions are the secant, tangent and missing cases of Result 1.
Exercises
Words
- True or false: every diameter is a chord. Explain.
- True or false: every chord is a diameter. Explain.
- A line meets a circle at exactly the points A and B. Is the line a chord, a secant or a tangent? What is \overline{AB}?
- Segment \overline{PQ} has P on a circle and Q at its center. Is it a chord? Explain.
- Can a tangent line contain a chord of the circle? Explain.
Distance and number of points
A circle has radius 8 cm. A line lies at the given distance from its center. How many points does it share with the circle, and what kind of line is it?
- 5 cm
- 8 cm
- 11 cm
- 0 cm. How long is the chord it cuts off?
- A line is tangent to a circle and lies 6.5 cm from its center. How long is the diameter of the circle?
- A circle has radius 10. A line lies 6 from its center. How long is the chord it cuts off?
Lines on a grid
A circle has center (-2, 3) and radius 5. For Exercises 12 to 15, say whether the line is a secant, a tangent or neither, and find the points it shares with the circle.
- y = 7
- x = 3
- y = -3
- x = -2
- Find every value of k for which the line y = k is tangent to this circle, and the point of tangency for each.
- For which values of k is the line x = k a secant of this circle?
Concentric circles
Two concentric circles have center O and radii 6 cm and 10 cm.
- A line lies 8 cm from O. How many points does it share with the two circles together?
- A line lies 6 cm from O. How many points does it share with the two circles together?
- A chord of the larger circle is tangent to the smaller one. How long is the chord?
- Two other concentric circles have an outer radius of 17. A chord of the outer circle is tangent to the inner circle and has length 30. Find the inner radius and the area of the ring between the circles.
Reasoning
- Explain why a line cannot meet a circle at three points.
- A point P is 4 cm from the center of a circle of radius 6 cm. Explain why no line through P is tangent to the circle.
- A line meets a circle at exactly one point T. Explain why T must be the point of the line nearest to the center.
- Explain why no line can be tangent to both of two different concentric circles.
- A line is tangent to a circle of radius r at T. Show that every point of the line other than T lies outside the circle.
Answers
- True. A diameter is a segment with both ends on the circle, which is what a chord is; it is the chord that happens to pass through the center.
- False. A chord that misses the center, such as \overline{PQ} in Figure 4, is not a diameter.
- The line is a secant, because it meets the circle at exactly two points. \overline{AB}, the part of it between those points, is a chord.
- No. It is a radius. A chord needs both ends on the circle, and the center Q is not on the circle.
- No. A chord has two endpoints on the circle, so a line containing it meets the circle at two points at least. A tangent meets the circle at only one.
- Two points, so the line is a secant: 5 < 8.
- One point, so the line is a tangent: the distance equals the radius.
- No points, so the line misses: 11 > 8.
- Two points, so the line is a secant through the center: 0 < 8. Its chord is the diameter, 2 \times 8 = 16 cm.
- 13 cm. A tangent lies at a distance equal to the radius, so r = 6.5 cm and the diameter is 2 \times 6.5 = 13 cm.
- 16. By Result 3 the chord is \displaystyle 2\sqrt{10^2 - 6^2} = 2\sqrt{64} = 16.
- A secant, meeting the circle at (-5, 7) and (1, 7). The distance from the center is 7 - 3 = 4 < 5; the foot is (-2, 7), and the points are \displaystyle \sqrt{25 - 16} = 3 to either side of it.
- A tangent at (3, 3). The distance is 3 - (-2) = 5, the radius, and the foot of the perpendicular from the center is (3, 3).
- Neither: it misses. The distance is 3 - (-3) = 6 > 5; a point on it would need FX^2 = 5^2 - 6^2 = -11.
- A secant through the center, meeting the circle at (-2, 8) and (-2, -2). Here d = 0, and the points are 5 above and below the center. Its chord is a diameter, of length 10.
- k = 8, touching at (-2, 8), and k = -2, touching at (-2, -2). Tangent means |k - 3| = 5, so k - 3 = 5 or k - 3 = -5.
- -7 < k < 3. The distance from the center to x = k is |k + 2|, and a secant needs |k + 2| < 5, so -5 < k + 2 < 5.
- 2. The inner circle: 8 > 6, so the line misses it. The outer circle: 8 < 10, so it crosses at two points.
- 3. The inner circle: 6 = 6, so the line is tangent to it, one point. The outer circle: 6 < 10, two points.
- 16 cm. The chord’s line is tangent to the inner circle, so it lies 6 cm from O. For the outer circle the half-chord is \displaystyle \sqrt{10^2 - 6^2} = 8, so the chord is 16 cm.
- The inner radius is 8, and the ring has area 225\pi. The half-chord is 15, and the line is at the inner radius r from the center, so r^2 + 15^2 = 17^2 and r^2 = 289 - 225 = 64. The ring’s area is \pi(17^2 - 8^2) = \pi \cdot 15^2 = 225\pi.
- Let F be the foot of the perpendicular from the center. A point X of the line is on the circle exactly when FX^2 = r^2 - d^2, so all such points are the same distance from F. On a line there are at most two points at one distance from a given point, one on each side, so the line and circle share at most two points.
- Any line through P is at most 4 cm from the center: the perpendicular from the center is the shortest segment from it to the line, and \overline{OP} is one such segment. A tangent would have to lie exactly 6 cm from the center, and 4 < 6, so every line through P is a secant.
- Let F be the point of the line nearest the center, so the line is at distance d = OF. If T were not F, then OF < OT = r, so d < r and by Result 2 the line would meet the circle at two points, not one. So T = F.
- A line has one distance d from the shared center. Being tangent to the inner circle needs d to equal the inner radius, and being tangent to the outer needs d to equal the outer radius. The radii are different, so both cannot hold.
- The line is tangent, so d = r and the foot of the perpendicular is T (Result 2). For any other point X on the line, OX^2 = r^2 + TX^2 with TX > 0, so OX^2 > r^2 and OX > r: X is outside the circle.