SSA can give two triangles
Show on a big screenTwo sides and an angle are given. The angle is not between them. A sweep is a compass arc drawn from a side's end. A sweep of 5 has radius 5.
- An angle of 30°, and a side of 8 next to it.
- A sweep of length 5, from the far end of the 8.
- The sweep crosses the slanted arm twice.
- Two different triangles come out, with third sides ≈3.9 and ≈9.9.
The rule
This arrangement is SSA: side, side, angle. The angle is not between the two sides.
The slanted arm is the other arm of the 30° angle. The sweep can cross it zero times, once or twice. Two crossings mean two different triangles from the same measurements. So SSA is not a rule that always works.
| Sweep | Crossings | Triangles |
|---|---|---|
| 3 | none | none |
| 4 | one | one |
| 5 | two | two |
| 9 | one | one |
Call the far end of the 8 point B. Draw the shortest line from B to the slanted arm. It meets the arm at 90°. That makes a right triangle with a 30° corner. Flip a copy over the arm. The two make an equilateral triangle of side 8. So the shortest line is half of 8, which is 4. A sweep shorter than 4 cannot reach the arm. A sweep of exactly 4 touches it once. A sweep between 4 and 8 crosses it twice. A sweep of 8 or more crosses it once.
Slide the sweep. Count the triangles.
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Try the lab to finish this lesson.