Loci
Two villages want to build a sports centre the same distance from both of them. A museum wants to put a display case the same distance from two walls. A dog tied to a 5 m lead can reach any point 5 m or less from the post. These are all problems about loci: the set of all the points that satisfy a condition.
What a locus is
The simplest one: the points that are at a distance $r$ from a point $O$ form the circle with centre $O$ and radius $r$. Check it for yourself: drag point P and leave "footprints" in the places where $OP$ is exactly the required length.
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…
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.
The perpendicular bisector
Look for points P with $PA = PB$. What shape do your footprints make?
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…
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 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…
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.
- a) Describe what locus is formed by all the points that are at the same distance from $A$ and from $B$.
- 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.
The angle bisector
The distance from a point to a line is always measured perpendicularly (it is the shortest path). In the interactive, the coloured lines are those perpendiculars:
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…
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.
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…
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.
- a) Define the locus that satisfies the first condition (being at the same distance from both streets).
- 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?
The special points of a triangle
In every triangle, certain lines always meet at a single point. Each point has its own use:
| Point | Where the… meet | Property | What it is used for |
|---|---|---|---|
| Circumcentre | Perpendicular bisectors | Equidistant from the 3 vertices | Centre of the circle that passes through the vertices (circumscribed circle). An antenna that gives equal coverage to three villages. |
| Incentre | Angle bisectors | Equidistant from the 3 sides | Centre of the inscribed circle (tangent to the sides). The fountain furthest from the fences of a triangular park. |
| Centroid | Medians (vertex → midpoint of the opposite side) | Divides each median in the ratio 2 : 1 | Centre of gravity: a triangular plate balances on it. |
| Orthocentre | Altitudes | — | — |
Identify the special point of a triangle described in each case: a) It is the point equidistant from the three vertices of the triangle. b) It is the point of…
Identify the special point of a triangle described in each case:
- a) It is the point equidistant from the three vertices of the triangle.
- b) It is the point of intersection of the three altitudes of a triangle.
- c) It is the centre of the inscribed circle of the triangle (the one that touches its three sides).
- d) It is the centre of gravity (centroid) of the triangle and the point of intersection of the medians.
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…
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.
- a) Which notable lines of the triangle would you have to draw to find the location?
- b) What is the resulting point called?
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…
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.
- a) What is this balance point called in geometry?
- 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…
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.
- a) Determine the coordinates of the point where the warehouse should be located by studying the perpendicular bisectors.
- 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?
Back to the beginning: The sports centre must be located on the perpendicular bisector of the segment joining the two villages, the display case on the bisector of the angle between the two walls, and the area the dog can reach is the disc of radius $5\text{ m}$ centred on the post.