Proving the 1:3 Ratio of Hypotenuse Segments in a 30°-60° Right Triangle

To prove that the altitude drawn to the hypotenuse of a 30°-60° right triangle divides the hypotenuse into segments whose lengths have the ratio 1:3, follow these detailed steps. This proof is fundamental in understanding geometric relationships and will help enhance your knowledge of triangular geometry.

Step 1: Define the Triangle

Consider a 30°-60° right triangle with vertices:

A: the 30° angle B: the 60° angle C: the right angle

Denote the side lengths as follows:

The side opposite the 30° angle, BC, is x. The side opposite the 60° angle, AC, is x√3. The hypotenuse AB will be 2x because in a 30°-60°-90° triangle, the hypotenuse is twice the length of the shorter leg.

Step 2: Find the Altitude

Let D be the foot of the altitude from vertex C to the hypotenuse AB. Denote the lengths of segments AD and DB on the hypotenuse AB.

Step 3: Calculate the Area in Two Ways

The area of triangle ABC can be calculated in two different ways:

Using the base and height: Area (1/2) × base × height (1/2) × AB × height (1/2) × 2x × h xh Using the legs: Area (1/2) × BC × AC (1/2) × x × x√3 (x2√3) / 2

Step 4: Equate the Areas

Setting the two area expressions equal to each other gives:

xh  (x2√3) / 2

Solving for the height h yields:

h  (x√3) / 2

Step 5: Use Similar Triangles

Triangles ACD and BCD are similar to triangle ABC:

Triangle ACD is a 30°-60° triangle. Triangle BCD is a 30°-60° triangle.

Let’s denote:

AD m DB n

From the properties of similar triangles, we can set up the following ratios:

h / m AC / AB h / n BC / AB

Substituting the known values:

( (x√3) / 2 ) / m (x√3) / (2x) → (√3 / 2m) (√3 / 2) → m 1 ( (x√3) / 2 ) / n x / (2x) → (√3 / 2n) (1 / 2) → n (√3 / 3)

Step 6: Find the Ratio

The hypotenuse AB AD DB m n 1 3 4. The segments AD and DB can be expressed as:

AD 1k DB 3k for some k

Thus, the ratio AD : DB 1 : 3.

Conclusion

Therefore, the altitude CD divides the hypotenuse AB into two segments AD and DB such that:

AD / DB  1 / 3

This confirms that the altitude on the hypotenuse divides it into segments whose lengths have the ratio 1:3. Understanding this proof is crucial for deeper knowledge of geometric properties in right triangles.