Chords, Secants, and Tangents
You're standing 20 ft from the center of a circular fountain with radius 12 ft. A taut rope runs from you to the edge of the fountain and just barely touches it, without cutting inside. How long is the rope? By the end of this page you'll solve it in one line, and you'll also know how to handle chords, secants, and even tangent lines to two circles at once.
Tangents and radii
Lines and segments around a circle have special names. Learn them first, because every theorem on this page uses them.
Look at the right-angle mark in the picture. A tangent and the radius drawn to the point of tangency always meet at 90°.
Why does it work? Every point of the tangent line except $T$ is outside the circle, so it is farther than $r$ from the center. That makes the radius $\overline{OT}$ the shortest path from $O$ to the line, and the shortest path to a line is always perpendicular to it.
Triangle $OTP$ is a right triangle with the right angle at $T$. The tangent segment $t$ and the radius $r$ are the legs, and the distance $d = OP$ from the center to the outside point is the hypotenuse. So the Pythagorean Theorem gives
$$t^2 + r^2 = d^2 \quad\Longrightarrow\quad t = \sqrt{d^2 - r^2}.$$If $PB = 10$, then $5^2 + 10^2 = 125 \ne 169$. There is no right angle, so the line is not tangent. Since $OB = 5$ is the radius, $B$ is on the circle, and the line is a secant through $B$.
A water tower is located $15 \text{ miles}$ away from the center of a circular city. A straight road is tangent to the city's boundary and passes through the water…
A water tower is located $15 \text{ miles}$ away from the center of a circular city. A straight road is tangent to the city's boundary and passes through the water tower. If the radius of the city is $9 \text{ miles}$, calculate the length of the road segment from the water tower to the point of tangency.
A weather balloon is at an altitude of $2 \text{ miles}$ above the surface of the Earth. Assuming the Earth is a perfect sphere with a radius of $3,959 \text{ miles}$,…
A weather balloon is at an altitude of $2 \text{ miles}$ above the surface of the Earth. Assuming the Earth is a perfect sphere with a radius of $3,959 \text{ miles}$, calculate the distance from the balloon to the horizon (the tangent segment) to the nearest tenth of a mile.
A civil engineer is verifying if a straight railway track is perfectly tangent to a circular safety zone. The radius of the safety zone is $150 \text{ ft}$. The distance…
A civil engineer is verifying if a straight railway track is perfectly tangent to a circular safety zone. The radius of the safety zone is $150 \text{ ft}$. The distance from the center of the zone to a signal box on the track is $250 \text{ ft}$. The distance along the track from the signal box to the point where it meets the boundary of the safety zone is $200 \text{ ft}$. Determine whether the track is perfectly tangent to the safety zone, intersects it at two points, or misses it entirely. Justify your conclusion using geometric properties.
Two tangents from one point
From a point outside a circle you can draw exactly two tangents. Look at the figure.
Why: $\overline{OT} \perp \overline{PT}$ and $\overline{OU} \perp \overline{PU}$, so triangles $OTP$ and $OUP$ are right triangles. They share the hypotenuse $\overline{OP}$, and the legs $\overline{OT}$ and $\overline{OU}$ are both radii. The triangles are congruent by HL, so $PT = PU$ by CPCTC.
The perimeter is $15 + 14 + 11 = 40$ ft. The shortcut: every tangent length appears twice, so the perimeter is $2(6 + 9 + 5) = 40$.
An observation deck $D$ is located outside a circular crater. Two straight walkways are built from $D$ to be perfectly tangent to the crater at points $T$ and $U$. If…
An observation deck $D$ is located outside a circular crater. Two straight walkways are built from $D$ to be perfectly tangent to the crater at points $T$ and $U$. If the walkway to point $T$ is $250 \text{ ft}$ long, determine the length of the walkway to point $U$.
A circular garden is completely enclosed by a triangular fence $ABC$ such that the fence is tangent to the garden at points $P$, $Q$, and $R$ on sides $AB$, $BC$, and…
A circular garden is completely enclosed by a triangular fence $ABC$ such that the fence is tangent to the garden at points $P$, $Q$, and $R$ on sides $AB$, $BC$, and $CA$ respectively. If $AP = 15 \text{ ft}$, $BQ = 20 \text{ ft}$, and $CR = 12 \text{ ft}$, calculate the total perimeter of the triangular fence.
A drone is hovering at a point outside a circular restricted airspace. The distance from the drone to the center of the airspace is $24 \text{ miles}$. The drone's…
A drone is hovering at a point outside a circular restricted airspace. The distance from the drone to the center of the airspace is $24 \text{ miles}$. The drone's camera determines that the angle between the two tangent lines to the airspace is $60^\circ$. What is the radius of the restricted airspace?
Chords and the center
A chord and the center of the circle form an isosceles triangle, since two of its sides are radii. That one fact gives all the chord theorems.
Here $c$ is the length of the whole chord, so each half measures $\tfrac{c}{2}$.
Why 1 and 2: triangle $OAB$ is isosceles with $OA = OB$, and in an isosceles triangle the altitude from the apex is also the median. Distance from a point to a chord always means the perpendicular distance.
The right triangle $OMA$ has legs $d$ and $\tfrac{c}{2}$ (half the chord) and hypotenuse $r$:
$$r^2 = d^2 + \left(\tfrac{c}{2}\right)^2$$Expanding: $r^2 = 64 + r^2 - 8r + 16$, so $8r = 80$ and $r = 10$ in.
In a circular swimming pool, two floating lane lines (chords) are installed. Lane 1 is $16 \text{ ft}$ long and is anchored $6 \text{ ft}$ from the center of the pool.…
In a circular swimming pool, two floating lane lines (chords) are installed. Lane 1 is $16 \text{ ft}$ long and is anchored $6 \text{ ft}$ from the center of the pool. If Lane 2 is also anchored $6 \text{ ft}$ from the center, determine the length of Lane 2.
A circular pizza has a radius of $10 \text{ inches}$. Two straight cuts are made to form chords of $16 \text{ inches}$ each. How far is each cut from the center of the…
A circular pizza has a radius of $10 \text{ inches}$. Two straight cuts are made to form chords of $16 \text{ inches}$ each. How far is each cut from the center of the pizza?
A chord of length $24 \text{ cm}$ is perpendicular to a diameter of a circle. The intersection point divides the diameter into two segments, one of which is…
A chord of length $24 \text{ cm}$ is perpendicular to a diameter of a circle. The intersection point divides the diameter into two segments, one of which is $8 \text{ cm}$ long. Find the radius of the circle.
Intersecting chords
When two chords cross inside a circle, each one is cut into two pieces. Something surprising happens when you multiply the pieces.
Why: triangles $AEC$ and $DEB$ are similar by AA. The angles at $E$ are vertical angles, and $\angle CAB \cong \angle CDB$ because both intercept the same arc $\widehat{CB}$ (you'll prove this in the inscribed angles section). So $\dfrac{AE}{DE} = \dfrac{CE}{BE}$. Cross-multiplying gives $AE \cdot BE = CE \cdot DE$.
$r = \sqrt{36.25} = \dfrac{\sqrt{145}}{2} \approx 6.02$.
Two straight hiking trails intersect inside a circular national park. Trail A is divided into two segments of $4 \text{ miles}$ and $6 \text{ miles}$ by the intersection…
Two straight hiking trails intersect inside a circular national park. Trail A is divided into two segments of $4 \text{ miles}$ and $6 \text{ miles}$ by the intersection point. Trail B is divided into two segments, one of which is $3 \text{ miles}$ long. Calculate the length of the other segment of Trail B.
Two support beams cross inside a circular stained-glass window. The first beam is split into segments of length $x$ and $x + 5$ inches. The second beam is split into…
Two support beams cross inside a circular stained-glass window. The first beam is split into segments of length $x$ and $x + 5$ inches. The second beam is split into segments of $x + 1$ and $x + 2$ inches. Find the value of $x$ and the total length of the longest beam.
Two straight underground pipes intersect at right angles at point $E$ inside a circular plaza. Pipe 1 is divided into segments of $6 \text{ ft}$ and $8 \text{ ft}$. Pipe…
Two straight underground pipes intersect at right angles at point $E$ inside a circular plaza. Pipe 1 is divided into segments of $6 \text{ ft}$ and $8 \text{ ft}$. Pipe 2 is divided into a $4 \text{ ft}$ segment and a $12 \text{ ft}$ segment. Calculate the exact radius of the circular plaza.
Two secants from a point
Now move the crossing point outside the circle. Two secants from the same point $P$ each cut the circle twice.
For each secant, the external part goes from $P$ to the near point, and the whole secant goes from $P$ to the far point.
Why: triangles $PAD$ and $PCB$ share $\angle P$ and have $\angle PDA \cong \angle PBC$ (both intercept arc $\widehat{AC}$), so they are similar by AA. Then $\dfrac{PA}{PC} = \dfrac{PD}{PB}$, which gives $PA \cdot PB = PC \cdot PD$.
$5x - 8x = -9$ gives $x = 3$. Check: $3 \cdot 8 = 24$ and $2 \cdot 12 = 24$. ✓
Two laser beams are fired from a single source $P$ outside a glass sphere. Beam 1 intersects the sphere at points $A$ and $B$, with the external segment…
Two laser beams are fired from a single source $P$ outside a glass sphere. Beam 1 intersects the sphere at points $A$ and $B$, with the external segment $PA = 5 \text{ in}$ and the internal chord $AB = 7 \text{ in}$. Beam 2 intersects the sphere at points $C$ and $D$, with the external segment $PC = 4 \text{ in}$. Find the length of the internal chord $CD$.
A new subway line is planned to cut through a circular city zone. From a station outside the zone, the distance to the near edge is $3 \text{ miles}$ and the distance…
A new subway line is planned to cut through a circular city zone. From a station outside the zone, the distance to the near edge is $3 \text{ miles}$ and the distance through the zone is $5 \text{ miles}$. A second proposed line from the same station reaches the near edge of the zone at $4 \text{ miles}$. How long is the path of this second line inside the zone?
A ship $S$ is located outside a circular atoll. Two straight radar lines are plotted from the ship through the atoll. The first line intersects the atoll at $A$ and $B$,…
A ship $S$ is located outside a circular atoll. Two straight radar lines are plotted from the ship through the atoll. The first line intersects the atoll at $A$ and $B$, with $SA = x$ and the internal segment $AB = 12 \text{ miles}$. The second line intersects at $C$ and $D$, with $SC = x - 2$ and the internal segment $CD = 18 \text{ miles}$. Find the distance from the ship to the near edge of the atoll along the first line (the value of $x$).
A tangent and a secant
What if one of the two lines from $P$ is a tangent? A tangent is like a secant whose two points have merged into one, so its external part and whole length are both $PT$.
Why: set $PC = PD = PT$ in the Two-Secant Theorem. Another way: triangles $PTA$ and $PBT$ are similar by AA.
When the secant passes through the center, $PA = d - r$ and $PB = d + r$. The product is $(d - r)(d + r) = d^2 - r^2$, exactly what the Pythagorean Theorem gave us in the first section. The theorems agree.
$r = 8$ in. Check: $d = 17$ and $17^2 - 8^2 = 289 - 64 = 225 = 15^2$. ✓
A spotlight is placed outside a circular stage. The beam forms a tangent line segment of $12 \text{ ft}$ to the edge of the stage. Another beam from the same spotlight…
A spotlight is placed outside a circular stage. The beam forms a tangent line segment of $12 \text{ ft}$ to the edge of the stage. Another beam from the same spotlight forms a secant that passes through the stage. If the external part of the secant (from the spotlight to the near edge of the stage) is $8 \text{ ft}$, what is the total length of the secant line from the spotlight to the far edge of the stage?
A straight highway is tangent to a circular nature reservation. A second dirt road starts from the same point on the highway, travels $4 \text{ miles}$ to reach the…
A straight highway is tangent to a circular nature reservation. A second dirt road starts from the same point on the highway, travels $4 \text{ miles}$ to reach the boundary of the reservation, and then goes straight through it. If the tangent segment is $10 \text{ miles}$ long, find the distance the dirt road travels inside the reservation.
From a point $P$ outside a circle, a tangent $PT$ and a secant $PAB$ are drawn. The length of the tangent is $20 \text{ inches}$. The secant passes through the center of…
From a point $P$ outside a circle, a tangent $PT$ and a secant $PAB$ are drawn. The length of the tangent is $20 \text{ inches}$. The secant passes through the center of the circle. If the external segment $PA$ is in a $1:3$ ratio with the diameter of the circle, find the radius of the circle.
Tangents in action
Three classic problems combine the ideas above. In each one the key move is the same: draw the radii to the points of tangency.
Constructing a tangent from an outside point
Given circle $O$ and an external point $P$, here is how to build a tangent. Draw $\overline{OP}$ and find its midpoint $M$. Draw the circle centered at $M$ with radius $MO$. It passes through both $O$ and $P$ and crosses circle $O$ at $A$ and $B$. Line $PA$ is tangent to circle $O$.
Why does it work? $\overline{OP}$ is a diameter of the circle centered at $M$, and $A$ is on that circle. An angle inscribed in a semicircle is a right angle (this is the US meaning of Thales' theorem; you'll prove it in the inscribed angles section). So $\angle OAP = 90^\circ$, meaning $\overline{PA} \perp \overline{OA}$ at the outer end of the radius $\overline{OA}$. By the converse of the Tangent–Radius Theorem, $PA$ is tangent to circle $O$ at $A$.
The circle inscribed in a right triangle
Take a right triangle with legs $a$ and $b$ and hypotenuse $c$, and a circle tangent to all three sides.
The radii to the two legs form a square with the right-angle vertex $C$, so the tangent segments from $C$ both equal $r$. By the Two-Tangent Theorem, the tangent segments from $A$ are both $x$ and from $B$ are both $y$. Then $a = r + y$, $b = r + x$ and $c = x + y$. Adding the legs and subtracting the hypotenuse:
$$a + b - c = 2r \quad\Longrightarrow\quad r = \frac{a + b - c}{2}.$$$r = \dfrac{30 + 40 - 50}{2} = 10$ ft. Check: $x = 40 - 10 = 30$ and $y = 30 - 10 = 20$, and $30 + 20 = 50$. ✓
A tangent line to two circles
Two circles with radii $R > r$ and centers $d$ apart have an external common tangent touching them at $T_1$ and $T_2$.
Both radii $\overline{O_1T_1}$ and $\overline{O_2T_2}$ are perpendicular to the tangent, so they are parallel. From $O_2$, draw a segment parallel to the tangent that meets $\overline{O_1T_1}$ at a point $Q$. Then $Q T_1 T_2 O_2$ is a rectangle, so $QT_1 = r$ and $QO_1 = R - r$. Triangle $O_1QO_2$ is a right triangle with hypotenuse $d$ and legs $R - r$ and the tangent length $t = T_1T_2$:
$$t^2 + (R - r)^2 = d^2 \quad\Longrightarrow\quad t = \sqrt{d^2 - (R - r)^2}.$$To construct a tangent to a circle $O$ from an external point $P$, a student first draws segment $OP$ and finds its midpoint $M$. Then, the student draws a circle…
To construct a tangent to a circle $O$ from an external point $P$, a student first draws segment $OP$ and finds its midpoint $M$. Then, the student draws a circle centered at $M$ with radius $OM$. The two circles intersect at points $A$ and $B$. Explain geometrically why drawing the line $PA$ guarantees that it is perfectly tangent to circle $O$ at point $A$.
Two circular gears in a machine have radii of $12 \text{ inches}$ and $7 \text{ inches}$. The distance between their centers is $13 \text{ inches}$. A straight drive…
Two circular gears in a machine have radii of $12 \text{ inches}$ and $7 \text{ inches}$. The distance between their centers is $13 \text{ inches}$. A straight drive belt is tangent to both gears externally. Find the length of the external tangent segment between the two points of tangency.
A right-angled triangular plot of land has legs measuring $30 \text{ ft}$ and $40 \text{ ft}$. A circular flower bed is constructed so that it is inscribed perfectly…
A right-angled triangular plot of land has legs measuring $30 \text{ ft}$ and $40 \text{ ft}$. A circular flower bed is constructed so that it is inscribed perfectly within the triangle, tangent to all three sides. What is the radius of the circular flower bed?
Back to the start: the rope to the fountain is a tangent segment, so it forms a right angle with the radius. That gives $\sqrt{20^2 - 12^2} = 16$ ft.