Exercises: Loci

A young man trains his dog in an open field. The dog is attached to an extendable leash of $5$ feet maximum length, anchored to a stake driven into the ground. Draw and describe the locus that represents the maximum area the dog can move in if it keeps the leash always taut.

Two towns, Town A and Town B, are located at a distance of $8$ miles from each other. The authorities want to build a bus stop that is exactly at the same distance from both towns. On which line or locus must they locate the stop? Represent it graphically in your notebook.

In a museum, two walls form an angle of $70^\circ$. A display case is to be installed so that it is always at the same distance from both walls. Draw the angle and trace the locus where the objects in the display case could be placed. What is the technical name for this line?

Identify the triangle center described in each case:

1. a) It is the point that is equidistant from the three vertices of the triangle.
2. b) It is the point where the three altitudes of a triangle intersect.
3. c) It is the center of the inscribed circle of the triangle (the one that touches its three sides).
4. d) It is the center of gravity of the triangle and the point where the medians intersect.

Draw a triangle in your notebook whose sides measure $6$ in, $8$ in and $10$ in. Draw its three perpendicular bisectors and mark the point where they intersect. Check, using a compass, that this point is the center of a circle that passes through the three vertices. What is the name of this point?

A park has a triangular shape. A fountain is to be installed so that the distance from the fountain to each of the three paths (sides of the park) is the same.

1. a) Which special lines of the triangle would you have to draw to find the location?
2. b) What is the resulting point called?

A new straight bike lane has been designed passing through the center of a city. For safety reasons, no buildings are allowed to be constructed within $15$ feet of this lane. Describe the locus that delimits the prohibited zone on both sides of the bike lane and draw a simple diagram.

Is it possible for the circumcenter, the incenter, the centroid, and the orthocenter to coincide at the same point in a triangle? If so, state what type of triangle it must be and draw it, marking this point.

In a school playground, two Wi-Fi antennas are located at points $A$ and $B$, separated by a distance of $12$ feet. A student is looking for a place to sit where they receive the same signal strength from both antennas.

1. a) Describe what locus is formed by all the points that are at the same distance from $A$ and from $B$.
2. b) If the student also wants to be exactly $10$ feet from antenna $A$, in how many possible points in the playground could they sit? Explain what geometric figures you would have to draw to find them.

A town council wants to install a street light in a small triangular plaza whose vertices are located at points $P$, $Q$ and $R$. They have decided that the street light must be at the same distance from the three corners (vertices) of the plaza so that the lighting is balanced.

1. a) What is the name of the notable point of the triangle where they must install the street light?
2. b) Which notable lines of the triangle is it necessary to draw to locate this point? Draw it schematically.

Two straight streets in a neighborhood, Main Street and Second Street, intersect forming an angle of $60^\circ$. We want to place a trash can that meets two conditions: it must be at the same distance from both streets and exactly $5$ feet from the intersection point of the streets.

1. a) Define the locus that meets the first condition.
2. b) Use the concept of locus to explain how you would determine the exact position of the trash can. How many possible solutions are there in total?

In a soccer practice session, a ball is placed at a point that is equidistant from the two sidelines (which are parallel to each other). If the width of the field is $60$ feet:

1. a) Describe the locus where the ball can be.
2. b) If the coach adds a restriction and says that the ball must be $20$ feet from a specific corner flag, draw a diagram of the situation and identify the points that meet both conditions.

A graphic designer is creating a logo based on a scalene triangle. They want to inscribe a circle inside the triangle so that it is tangent to all three sides.

1. a) What notable point of the triangle must be the center of this circle?
2. b) If the distance from this center to one of the sides is $3$ in, what will be the distance to the other two sides? Justify your answer based on the definition of this locus.

Three friends, Ashley, Brian, and Carla, are located in a park forming a triangle. Ashley and Brian are $10$ feet apart. Carla observes that she is at a point such that the angle formed by her lines of sight to Ashley and Brian is always $90^\circ$.

1. a) What is the name of the locus where Carla is located?
2. b) If Carla moves while always maintaining this angle of $90^\circ$ with respect to her friends, what geometric figure would her trajectory describe?

A board game has been designed where a counter must be placed at the "center of gravity" of a triangular piece of cardboard so that it balances on the point of a pencil. The sides of the triangle measure $15$ in, $20$ in and $25$ in.

1. a) What name is given to this balance point in geometry?
2. b) If we draw one of the medians of this triangle, in what proportion does the balance point divide this median? Express the relationship between the distance from the point to the vertex and the distance from the point to the opposite side.

A logistics company wants to locate a central warehouse to serve three stores located at points $A(2, 6)$, $B(2, 0)$ and $C(10, 0)$ (units in miles). To minimize transport costs, the warehouse must be at the same distance from all three stores.

1. a) Determine the coordinates of the point where the warehouse must be located by studying the perpendicular bisectors.
2. b) What is the name of the locus of points that are at a distance of $5$ miles from point $B$? Would store $C$ be inside that zone?

A mountain rescue team receives signals from two lost hikers, located at points $A$ and $B$. The command center knows that the hikers are exactly at the same distance from each other, but the rescue helicopter must follow a straight trajectory represented by a secondary road $r$. If on a coordinate map the points are $A(1, 6)$ and $B(5, 2)$, and the road follows the direction of the line passing through $(0, 0)$ and $(8, 8)$, determine by means of locus reasoning the exact point where the helicopter will cross the line where the hikers could be found.

A telephone antenna is to be installed to serve three villages located at the vertices of a triangle whose sides measure $13\text{ mi}$, $14\text{ mi}$ and $15\text{ mi}$. The chief engineer decides that the antenna must be at the circumcenter of the triangle so that the distance to the three villages is the same. If the area of the triangle is $84\text{ mi}^2$, calculate the coverage radius the antenna must have (the distance from the circumcenter to any of the vertices) knowing that in any triangle the radius of the circumscribed circle satisfies the relationship $R = \frac{a \cdot b \cdot c}{4 \cdot \text{Area}}$.

In an amusement park, a new "suspended flight" ride follows a trajectory such that the sum of its distances to two fixed control towers, $F_1$ and $F_2$, is always $50\text{ meters}$. The towers are separated by a distance of $30\text{ meters}$.

1. Identify which locus this is.

2. If we place the center of the trajectory at the origin $(0,0)$, what will be the maximum height the ride will reach with respect to the axis joining the two towers? (Hint: use the Pythagorean theorem with the semi-axes of the figure).

An architect designs a triangular plaza whose sides measure $30$, $40$ and $50$ feet. Inside, he wants to place a fountain that is at the same distance from the three streets (the three sides of the triangle).

1. What is the name of the locus that determines the position of the fountain?

2. Calculate the distance from the fountain to each of the streets. (Hint: The area of a triangle can also be calculated as the product of the semiperimeter and the radius of the inscribed circle, $A = s \cdot r$).

Consider two intersecting lines $r$ and $s$ that cross forming an angle of $60^\circ$. We want to draw a circle of radius $10\text{ cm}$ that is tangent to both lines and lies inside the $60^\circ$ angle shown in the figure.

1. Explain on which line the center of this circle must be located.

2. Calculate at what distance from the intersection point of the two lines the center of the circle must be placed. (Hint: Joining the center to the point of tangency forms a right triangle; use the properties of triangles with angles of $30^\circ$ and $60^\circ$).

The "arc capable" of an angle $\alpha$ on a segment $AB$ is the locus of points from which the segment subtends the same angle $\alpha$. If we have a segment $AB$ of length $8\text{ cm}$, draw and describe the locus for an angle of $90^\circ$. Justify, based on the properties of inscribed angles, why this locus is a circle of diameter $AB$ and calculate its area.

A lighthouse emits a signal that is only received at a distance of less than $10\text{ mi}$. A boat moves along a straight coastline $c$ which is at a distance of $8\text{ mi}$ from the lighthouse at its closest point.

1. Describe the locus of the points on the coast that receive the lighthouse signal.

2. Calculate the length of the stretch of coast where the boat will be within the range of the lighthouse.

In an isosceles triangle $ABC$, with $AB = AC$, we know that the centroid (point where the medians intersect), the orthocenter (point where the altitudes intersect) and the circumcenter (point where the perpendicular bisectors intersect) are all aligned on the same straight line.

1. What is the name of this straight line that contains these notable points?

2. If the base $BC$ measures $12\text{ cm}$ and the altitude corresponding to that base measures $9\text{ cm}$, determine the position of the centroid with respect to the base, justifying your answer with the property of the locus of the medians.