Exercises: Loci

A young person is training their 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 formed by the positions through which the dog can move if it keeps the leash always taut (that is, stretched to the maximum). Choose a suitable scale, for example $1$ cm in the drawing equals $1$ m in real life, and mark the stake and the position of the dog in your drawing.

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

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 centre of the inscribed circle of the triangle (the one that touches its three sides).
4. d) It is the centre of gravity (centroid) of the triangle and the point of intersection of the medians.

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

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

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 has been designed that passes through the center of a city. For safety reasons, no building is allowed to be constructed less than $15$ meters from this lane. Describe the locus that bounds the forbidden zone on both sides of the bike lane and draw a simple sketch.

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

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 3rd year of ESO class, the teacher asks to define and draw the locus of points that are equidistant from two posts, $A$ and $B$, separated by a distance of $6\text{ m}$. Sergio writes in his notebook:\n\n“The locus is the segment of length $6\text{ m}$ that passes right through the midpoint between $A$ and $B$ and is perpendicular to the line joining them; if we go beyond those $6\text{ m}$, the points can no longer be equidistant from $A$ and $B$ because they would be too far away”.\n\na) Identify the two conceptual errors Sergio makes in his reasoning.\nb) Explain why the distance at which a point is located does not prevent it from belonging to said locus, and write down the correct definition of the locus in question.

On a sports court, two training markers are located at points $A$ and $B$, separated by exactly $8\text{ m}$. We want to place a ball in a position that satisfies two conditions simultaneously: being equidistant from $A$ and $B$, and being exactly $5\text{ m}$ from marker $A$.\n\na) Estimate: before making precise calculations or drawings, provide a justified estimate of how many possible positions exist on the court to place the ball (none, one, two, or infinitely many?). Support your prediction by intuitively comparing the search radius with the distance between the markers.\nb) Verify: formally identify which two loci correspond to each condition. Then, calculate the exact distance from the ball to the straight line joining $A$ and $B$ by applying the Pythagorean theorem to the corresponding triangle, and conclude whether your initial estimate was correct.

In a school playground, two Wi-Fi antennas are located at points $A$ and $B$, separated by a distance of $12$ meters (the figure shows only the antennas). 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 $B$.

1. 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 have to draw to find them.

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

1. a) What is the name of the triangle's notable point where they should install the street lamp?

1. b) Which notable lines of the triangle do you need to draw to locate that point? Draw it schematically.

Two straight streets in a neighborhood, Calle Mayor and Calle Menor, cross at a point $O$ forming an angle of $60^\circ$, as shown in the figure. You want to place a trash bin that meets two conditions: it must be at the same distance from both streets and it must be exactly $5$ meters from the intersection point $O$. Consider the streets as lines that extend indefinitely in both directions.

1. a) Define the locus that satisfies the first condition (being at the same distance from both streets).

1. b) Use the concept of locus to explain how you would determine the exact position of the trash bin. How many possible solutions are there in total?

During a football 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 pitch is $60$ metres:

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 $20$ metres from a specific corner flag, draw a diagram of the situation and identify the points that satisfy both conditions.

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

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 centre of this circle?
2. b) If the distance from that centre 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 standing 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 the lines of sight to Alba and Bruno is always $90^\circ$. The figure shows one of Carla's possible positions.

1. a) What is the locus where Carla is located called?
2. b) If Carla moves while always keeping that $90^\circ$ angle with respect to her friends, what geometric figure would her path describe? State where its center is and what its radius measures.

A board game has been designed in which a token must be placed at the "center of gravity" of a triangular cardboard piece so that it stays balanced on the tip of a pencil. The sides of the triangle measure $15$ cm, $20$ cm and $25$ cm. In the figure you can see the piece $ABC$ (with the right angle at $A$), its three medians and the point $P$ where the token is placed.

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?

The emergency services of Castilla y León are studying where to locate a regional ambulance base that responds with identical response times to the town centres of three municipalities: Arévalo ($A$), Medina del Campo ($M$), and Olmedo ($O$). The straight-line distances between their centres are approximately: $AM = 34\text{ km}$, $AO = 22\text{ km}$, and $MO = 20\text{ km}$. It is decided to locate the base at a point strictly equidistant from the three municipalities.\n\na) Model the situation: define which locus determines the points equidistant from any two municipalities and which triangle centre of the triangle formed by $A$, $M$, and $O$ represents the location of the base.\nb) Apply the relationship between the sides of a triangle ($c^2$ compared to $a^2 + b^2$) to classify triangle $AMO$ as acute, right, or obtuse.\nc) Based on the previous classification, reason whether the base will lie inside the triangle bounded by the three towns or outside it. Explain what practical consequence this property of loci has for territorial management.

In a Cartesian urban planning map, it is desired that the service boundary between two repeaters, $A$ and $B$ (that is, the perpendicular bisector of segment $AB$), be exactly the bisector line of the first and third quadrants, whose equation is $y = x$.\n\na) If repeater $A$ is installed at the coordinates $(0, 8)$, deduce in a reasoned way the coordinates where repeater $B$ must necessarily be placed. Check your deduction by verifying that the midpoint of segment $AB$ belongs to the line $y = x$ and that segment $AB$ is perpendicular to it.\nb) Generalise: formulate what algebraic relationship must exist between the coordinates $(x_A, y_A)$ and $(x_B, y_B)$ of any pair of points so that their perpendicular bisector remains the line $y = x$. Propose two other distinct pairs of points that share that same perpendicular bisector.

A mountain rescue team receives signals from two lost hikers, located at points $A$ and $B$. The command center knows that the search area is made up of the points that are exactly the same distance from $A$ and from $B$. The rescue helicopter must follow a straight path represented by a secondary road $r$. On a coordinate map, the points are $A(1, 6)$ and $B(5, 2)$, and the road follows the direction of the line that passes through $(0, 0)$ and $(4, 8)$.

Using loci reasoning, determine the exact point where the helicopter will cross the line where the hikers could be found.

It is desired to install a mobile phone mast 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 mast must be at the circumcentre 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 mast must have (the distance from the circumcentre to any of the vertices), given 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 moves so that it stays always at the same distance from two fixed control towers, $T_1$ and $T_2$, which are $30$ meters apart.

1. Identify which locus this is and describe it.

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

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

An architect designs a triangular square whose sides measure $30$, $40$, and $50$ metres. 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, $A = s \cdot r$).

Consider two intersecting lines $r$ and $s$ that meet at a point $V$ 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. a) Explain on which line the center $O$ of this circle must lie.

1. b) Calculate the distance from the intersection point $V$ of the two lines at which 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 $60^\circ$.)

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.

An agricultural cooperative purchases a triangular plot of land with corners at the intersections of three straight local roads. The roads form a right-angled triangle whose sides measure $60\text{ m}$, $80\text{ m}$, and $100\text{ m}$. A community well needs to be built inside the plot, and two geometric methods based on loci are debated:\n\n- Method 1: locate the well at the circumcentre of the plot (intersection of the perpendicular bisectors), with the argument that it must be equidistant from the three vertices (road intersections).\n- Method 2: locate the well at the incentre of the plot (intersection of the interior angle bisectors), with the argument that it must be equidistant from the three roads (sides of the plot).\n\nTo ensure the safety of machinery and prevent spills, agricultural regulations require that the well be located strictly within the interior of the land and at the greatest possible distance from the edge of any road.\n\na) Analyse the feasibility of both methods by calculating the exact distance at which the well would be located from each of the three roads in Method 1 and in Method 2 (you can calculate the radius of the incircle by equating the area of the triangle, $A = \frac{b \cdot h}{2}$, to the sum of the areas of the three triangles formed from the incentre, $A = \frac{a+b+c}{2} \cdot r$).\nb) Critically justify which of the two methods should be chosen and why the other is completely inadmissible according to the required regulations.

Salvamento Marítimo detects a distress signal in the vicinity of the Strait of Gibraltar. Two coastal stations located in Tarifa ($T$) and Algeciras ($A$), separated by $20\text{ km}$ in a straight line, coordinate the search based on three technical pieces of data:\n\n1. The intensity of the received signal is identical at both coastal stations.\n2. The range of the rescue vessel based in Tarifa allows it to cover a maximum search radius of $15\text{ km}$ from that station.\n3. To avoid reefs and the dense coastal navigation channel, any position whose distance to the straight line joining Tarifa and Algeciras is less than or equal to $6\text{ km}$ is ruled out.\n\na) Mathematically model the problem by translating each of the three conditions into the language of loci (perpendicular bisector, circumference/circle, or parallel distance bands to a line).\nb) Rigorously deduce the geometric shape of the zone where the vessel may be located: determine whether it is a two-dimensional region or a set of segments, and calculate the minimum and maximum distance to the straight line $TA$ of the points forming said zone.\nc) If a sonar buoy confirms that the exact distance from the vessel to Tarifa is $12,5\text{ km}$, calculate how many possible locations remain for the vessel and at what exact distance from the straight line $TA$ each of them is located.