Exercises: Loci

A young man is training his dog in an open field. The dog is attached to an extendable leash with a maximum length of $5$ meters, anchored to a stake driven into the ground. Draw and describe the locus representing the maximum boundary along which the dog can move if it keeps the leash fully extended.

Two villages, Village A and Village B, are located $8$ km apart, as shown in the figure. The authorities want to build a bus stop that is exactly the same distance from both villages. On which line or locus should they place the stop? Represent it graphically in your notebook.

In a museum, two walls form an angle of $70^\circ$. You want to install a display case that is always located 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 this line technically called?

You can do it on paper and send a photo with your answer.

Identify the special point of a triangle described in each case:

1. a) It is the point equidistant from the three vertices of the triangle.
2. b) It is the point of intersection of the three altitudes of a triangle.
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 (centroid) of the triangle and the point of intersection of the medians.

Draw a triangle in your notebook whose sides measure $6$ cm, $8$ cm, and $10$ cm. Construct its three perpendicular bisectors and mark the point of intersection. Using a compass, check that this point is the center of a circle passing through the three vertices. What is this point called?

A park has a triangular shape, as shown in the figure. You want to install a fountain inside the park so that the distance from the fountain to each of the three paths (sides of the park) is the same.

1. a) Which notable lines of the triangle would you have to draw to find the location?

1. b) What is the resulting point called?

A new straight bike lane passing through a city center has been designed. For safety reasons, no buildings are allowed within $15$ meters of this lane. Describe the locus bounding the restricted zone on both sides of the bike lane and draw a simple sketch.

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

In a school playground, two Wi-Fi antennas are located at points $A$ and $B$, separated by a distance of $12$ meters. 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 equidistant from $A$ and $B$.
2. b) If the student also wants to be exactly $10$ meters from antenna $A$, at how many possible points in the playground could they sit? Explain which geometric figures you would need to draw to find them.

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

1. a) What is the name of the special point of the triangle where the lamppost should be installed?
2. b) Which special lines of the triangle need to be drawn to locate this point? Draw a rough sketch.

Two straight streets in a neighborhood, Calle Mayor and Calle Menor, intersect forming an angle of $60^\circ$. A trash can is to be installed satisfying two conditions: it must be at the same distance from both streets and it must be exactly $5$ meters from the point of intersection of the streets.

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

During a soccer training session, a ball is placed at a point that is equidistant from the two touchlines (which are parallel to each other). If the width of the field is $60$ meters:

1. a) Describe the locus of points where the ball could be.
2. b) If the coach adds a constraint and says that the ball must be $20$ meters from a specific corner flag, draw a diagram of the situation and identify the points that satisfy 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) Which special point of the triangle must be the center of this circle?
2. b) If the distance from that center to one of the sides is $3$ cm, what will be the distance to the other two sides? Justify your answer based on the definition of that locus.

Three friends, Alba, Bruno, and Carla, are in a park forming a triangle. Alba and Bruno are $10$ meters apart. Carla notices that she is at a point such that the angle formed by her lines of sight to Alba and Bruno 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 that angle of $90^\circ$ relative to her friends, what geometric shape would her path describe?

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

1. a) What is this balance point called in geometry?
2. b) If we draw one of the medians of this triangle, in what ratio does the balance point divide that 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 that serves three stores located at points $A(2, 6)$, $B(2, 0)$ and $C(10, 0)$ (units in km). 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 should be located by studying the perpendicular bisectors.

1. b) What is the locus of points that are at a distance of $5$ km from point $B$ called? Would store $C$ be inside that area?

A mountain rescue team receives signals from two lost hikers, located at points $A$ and $B$. The command center knows that the hikers are at the exact same distance from each other, but the rescue helicopter must follow a straight trajectory represented by a minor 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 using locus reasoning the exact point where the helicopter will cross the line where the hikers could be located.

It is desired to install a cell phone tower to serve three villages located at the vertices of a triangle whose sides measure $13\text{ km}$, $14\text{ km}$, and $15\text{ km}$. The chief engineer decides that the tower 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 $84\text{ km}^2$, calculate the coverage radius the tower must have (the distance from the circumcenter to any of the vertices), given that in any triangle the radius of the circumscribed circle satisfies the relationship $R = \frac{a \times b \times c}{4 \times \text{Area}}$.

In a theme park, a new "suspended flight" ride moves in such a way that it remains always at the same distance from two fixed control towers, $T_1$ and $T_2$, separated from each other by $30$ meters.

1. Identify which locus this is and describe it.

2. If we place the towers at points $T_1(-15, 0)$ and $T_2(15, 0)$ on a plane (units in meters), what line describes the path of the ride? Give its equation.

3. The ride passes directly over a maintenance hut located at point $P(0, 20)$. Verify that the hut is equidistant from both towers by calculating both distances.

An architect designs a triangular plaza whose sides measure $30$, $40$, and $50$ meters. Inside, they want 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 semi-perimeter and the radius of the inscribed circle, $A = s \times r$).

Consider two intersecting lines $r$ and $s$ that meet at 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 which line the center of this circle must lie on.

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

Two rescue buoys, $A$ and $B$, float in the sea 8 km apart. The rescue services want to install a platform that meets two conditions at the same time: it must be equidistant from both buoys and it must be exactly $5$ km from buoy $A$.

On the map in the figure, with units in km, the buoys are at $A(0, 0)$ and $B(8, 0)$.

1. a) What locus do the points that are equidistant from $A$ and $B$ form? And the points that are $5$ km from $A$?

1. b) At how many different points can the platform be installed? Justify your answer by reasoning about the relative position of the two loci above.

1. c) Calculate the coordinates of all the points where the platform can be installed. (Hint: use the Pythagorean theorem.)

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

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

2. Calculate the length of the stretch of coastline along which the boat will be within range of the lighthouse.

In an isosceles triangle $ABC$, with $AB = AC$, we know that the centroid (point of intersection of the medians), the orthocentre (point of intersection of the altitudes), and the circumcentre (point of intersection of the perpendicular bisectors) are all aligned along the same straight line.

1. That line coincides with the median, altitude, perpendicular bisector, and angle bisector drawn from vertex $A$. How is it related to the axis of symmetry of the triangle? Justify your answer.

2. If the base $BC$ measures $12\text{ cm}$ and the altitude corresponding to that base measures $9\text{ cm}$, determine how far from the base the centroid is located, knowing that the centroid divides each median into two segments such that the one reaching the vertex is twice as long as the other.