Constructing a Triangle Given Three Elements
Geometry โ Grade 7 | Compass and Straightedge Constructions
๐ Introduction: What Are Construction Problems?
In geometry, a construction problem is a task where we build geometric figures using only two tools:
๐ต Compass
Used to draw circles and arcs, and to transfer distances (lengths) from one place to another.
๐ Straightedge
Used to draw straight lines through two points. It has no markings โ you cannot measure with it!
Key Idea: A triangle has six elements โ three sides and three angles. It turns out that knowing just three specific elements is enough to completely determine a triangle (up to congruence).
The Three Classical Construction Tasks
| Task | Given Elements | Abbreviation |
|---|---|---|
| Task 1 | Two sides and the included angle | SAS |
| Task 2 | One side and two adjacent angles | ASA |
| Task 3 | Three sides | SSS |
๐บ Task 1: Two Sides and the Included Angle (SAS)
Given:
- Segment $P_1Q_1$ (representing side $AB$)
- Segment $P_2Q_2$ (representing side $AC$)
- Angle $\angle hk$ (the included angle at vertex $A$)
Required:
Construct $\triangle ABC$ such that $AB = P_1Q_1$, $AC = P_2Q_2$, and $\angle BAC = \angle hk$.
โ ๏ธ Important: The angle must be the included angle โ the angle between the two given sides. Using the wrong angle is a common error!
Construction Steps:
Step 1: Draw a line $a$. On this line, mark a point $A$. Using the compass, measure the length $P_1Q_1$ and mark point $B$ on line $a$ so that $AB = P_1Q_1$.
Step 2: At point $A$, construct an angle $\angle BAM$ equal to the given angle $\angle hk$. (This creates a ray $AM$ from $A$.)
Step 3: On ray $AM$, use the compass to mark point $C$ such that $AC = P_2Q_2$.
Step 4: Connect points $B$ and $C$ with a straight line.
โ
Result: $\triangle ABC$ is the required triangle with $AB = P_1Q_1$, $AC = P_2Q_2$, and $\angle BAC = \angle hk$.
Interactive Diagram โ SAS Construction:
Interactive visualization โ available in the full lesson
๐ Uniqueness (SAS): There are infinitely many triangles that satisfy these conditions (you can draw the triangle anywhere on the plane), but they are all congruent to each other by the SAS Congruence Criterion. So the solution is essentially unique.