Solving a 30-60-90 Triangle: Finding Side Lengths with Given Hypotenuse

In this article, we will explore how to find the lengths of the sides in a right triangle with given angle measures. Specifically, we will solve for AB and BC in triangle ABC, where A 30°, B 90°, C 60°, and the hypotenuse AC 16 cm. We will use properties of 30-60-90 triangles and trigonometric principles to arrive at our solution.

Properties of the 30-60-90 Triangle

A 30-60-90 triangle is a special right triangle where the angles are 30°, 60°, and 90°. The sides of such a triangle are in a fixed ratio of 1 : √3 : 2. Here, the side opposite the 30° angle is half the hypotenuse, and the side opposite the 60° angle is (√3)/2 of the hypotenuse.

Step-by-Step Solution

Step 1: Identify the Triangle Type

Since angle B is 90°, triangle ABC is a right triangle. We can use the properties of a 30-60-90 triangle to find the lengths of the other sides.

Step 2: Use the Properties of a 30-60-90 Triangle

The side opposite the 30° angle, which is AB, is 1/2 of the hypotenuse which is AC. The side opposite the 60° angle, which is BC, is (√3)/2 of the hypotenuse.

Step 3: Calculate the Lengths

Let's calculate the lengths of AB and BC using the given hypotenuse AC 16 cm.

Calculating AB

Using the formula: AB (1/2) × AC $$ AB frac{1}{2} times 16 , text{cm} 8 , text{cm} $$

Calculating BC

Using the formula: BC (√3/2) × AC $$ BC frac{sqrt{3}}{2} times 16 , text{cm} 8sqrt{3} , text{cm} approx 13.86 , text{cm} $$

Summary of Results

The lengths of the sides are as follows: AB: 8 cm BC: (8sqrt{3}) cm, approximately 13.86 cm This method allows us to find the lengths of the sides in a 30-60-90 triangle given the hypotenuse.

Alternative Methods Using Trigonometry

Using the sine rule, we can also calculate the side lengths as follows:

Using Sine Rule

The sine rule states that in any triangle:

Given that A 30°, B 90°, and C 60°, we can substitute the angles and the given hypotenuse into the formula.

For AB: $$frac{AB}{sin 60°} frac{16}{sin 90°}Rightarrow AB 16 times frac{sin 60°}{sin 90°} 16 times frac{frac{sqrt{3}}{2}}{1} 8sqrt{3} , text{cm} approx 13.86 , text{cm}$$

For BC: $$frac{BC}{sin 30°} frac{16}{sin 90°}Rightarrow BC 16 times frac{sin 30°}{sin 90°} 16 times frac{frac{1}{2}}{1} 8 , text{cm}$$

Trigonometric Calculation for Confirmation

Using trigonometric principles, we can also calculate the side lengths as follows:

Using Sine Function

For AB: $$sin 60° frac{AB}{AC} frac{sqrt{3}/2}{1} Rightarrow AB sin 60° times 16 frac{sqrt{3}}{2} times 16 8sqrt{3} , text{cm} approx 13.86 , text{cm}$$

For BC: $$sin 30° frac{BC}{AC} frac{1/2}{1} Rightarrow BC sin 30° times 16 frac{1}{2} times 16 8 , text{cm}$$

Conclusion

In summary, we have solved for the side lengths of triangle ABC using the properties of a 30-60-90 triangle and trigonometric principles. The lengths of AB and BC are 8 cm and 13.86 cm, respectively. This solution is useful in various geometric and trigonometric contexts.