How do you find the side lengths for a right triangle where only the angles of 90, 18, and 72 degrees are given?

Question

Carter Anderson · Accepted Answer

You can't.  There are infinitely many similar triangles having those angle measurements. (Recall from geometry that AAA does not establish congruence of triangles, but only similarity.)

Madelyn Adams · Answer

undefined To find the side lengths of a right triangle when only the angles of $90^\circ$, $18^\circ$, and $72^\circ$ are given, follow these steps:

1. Recognize that the triangle must be a special type known as a "golden triangle" or a "golden right triangle."
2. In a golden triangle, the ratio of the lengths of the sides follows the golden ratio, $φ$, which is approximately $1.618$.
3. The side lengths can be found using trigonometric functions and properties of similar triangles.
4. Label the sides of the triangle as follows:
   - The side opposite the $90^\circ$ angle is the hypotenuse (H).
   - The side opposite the $72^\circ$ angle is the base (B).
   - The side opposite the $18^\circ$ angle is the height (h).
5. Since the sum of the angles in a triangle is $180^\circ$, the remaining angle is $180^\circ - 90^\circ - 18^\circ = 72^\circ$.
6. Using trigonometric ratios, express the ratios of the sides of the triangle:
   - $ \frac{B}{H} = \sin(18^\circ) $ (by definition of sine)
   - $ \frac{h}{H} = \sin(72^\circ) $
   - $ \frac{h}{B} = \tan(18^\circ) $ (by definition of tangent)
7. Solve these equations to find the side lengths. Since the golden ratio is $ \frac{B}{h} = φ $, you can use this ratio to find one side length if another is known.
8. Alternatively, you can use geometric constructions or trigonometric relationships to determine the side lengths more precisely.

In summary, to find the side lengths of a right triangle with angles $90^\circ$, $18^\circ$, and $72^\circ$, you can use trigonometric ratios and properties of similar triangles, with specific consideration of the golden ratio.