Geometric Loci
Two towns in Oaxaca want to build a sports complex the same distance from both of them. A museum wants to place a display case the same distance from two walls. A dog tied to a 5 m leash can reach any point 5 m or less from the stake. These are all problems about geometric loci: the set of all points that meet a condition. You already drew the perpendicular bisector and the angle bisector with a straightedge and compass in Year 1; here you will see them as geometric loci and use them to find the notable points of a triangle.
What Is a Geometric Locus
The simplest one: the points that are a distance $r$ from a point $O$ form the circle with center $O$ and radius $r$. Check it yourself: drag point P and leave "footprints" at the places where $OP$ equals exactly the required distance.
A young man is training his dog in an open field. The dog is attached to a retractable leash with a maximum length of $5$ meters, anchored to a stake driven into the…
A young man is training his dog in an open field. The dog is attached to a retractable 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.
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. A student is looking for a place to sit where they…
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.
- a) Describe what locus is formed by all the points that are equidistant from $A$ and $B$.
- 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 need 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 colored 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, Avenida Juárez and Calle Hidalgo, intersect forming an angle of $60^\circ$. A trash can is to be installed satisfying two…
Two straight streets in a neighborhood, Avenida Juárez and Calle Hidalgo, 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.
- a) Define the locus that satisfies the first condition.
- 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?
The Notable Points of a Triangle
In every triangle, certain lines always meet at a single point. Each point has a use:
| Point | Where these meet… | Property | What it is used for |
|---|---|---|---|
| Circumcenter | Perpendicular bisectors | Equidistant from the 3 vertices | Center of the circle that passes through the vertices (circumscribed circle). An antenna that gives equal coverage to three towns. |
| Incenter | Angle bisectors | Equidistant from the 3 sides | Center of the inscribed circle (tangent to the sides). The fountain farthest from the fences of a triangular park. |
| Centroid | Medians (vertex → midpoint of the opposite side) | Divides each median in a 2 : 1 ratio | Center of gravity: a triangular plate balances on it. |
| Orthocenter | 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 center of the inscribed circle of the triangle (the one that touches its three sides).
- d) It is the center 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 counter must be placed at the "center of gravity" of a triangular cardboard piece so that it balances on the tip of a pencil.…
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.
- 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?