Exercises: Chords, secants and tangents

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.

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.

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?

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.

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 $CD$.

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 surveyor is standing at point $S$, looking at a circular building with center $C$. The line of sight to the edge of the building is tangent to the circle at point $T$. If the angle $\angle SCT$ measures $54^\circ$, what is the measure of $\angle CST$?

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 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.

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.

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.

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 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?

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.

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 $A$.

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.

From a point $P$ outside a circle, a tangent segment $PT$ is drawn measuring $12 \text{ inches}$. A secant line from $P$ passes through the center of the circle, intersecting it at points $A$ and $B$ (with $A$ closer to $P$). If $PA = 8 \text{ inches}$, find the radius of the circle. Then, calculate the length of a chord $CD$ that is perpendicular to $AB$ and passes through the midpoint of segment $OA$.

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.

A circular park features a large circular fountain in the center. A straight pedestrian bridge is built over the park. The bridge forms a chord of the outer boundary of the park and is perfectly tangent to the inner fountain. If the bridge is $400 \text{ ft}$ long, calculate the exact area of the walking space (the annulus) between the fountain and the park's outer boundary.

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 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 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?

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.

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.