Problem 37 - Elementary and Intermediate Algebra

Question

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the length of the leg opposite the \(60^{\circ}\) angle is 55 millimeters. Find the length of the leg opposite the \(30^{\circ}\) angle and the length of the hypotenuse. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

The leg opposite 30° is \( \frac{55 \sqrt{3}}{3} \) mm (~31.81 mm) and the hypotenuse is \( \frac{110 \sqrt{3}}{3} \) mm (~63.62 mm).

Step 1: Understanding the Triangle's Properties

A 30°-60°-90° triangle has sides in a specific ratio: for the angles 30°, 60°, and 90°, the ratio of the side lengths is 1 : \( \sqrt{3} \) : 2, respectively. In these triangles, the side opposite 30° is the shortest leg, the side opposite 60° is the longer leg, and the hypotenuse is opposite the 90° angle.

Step 2: Relate the Given Length to the Triangle's Properties

We are given that the length of the leg opposite the 60° angle is 55 mm, which corresponds to \( \sqrt{3} \times x \), where \( x \) is the length of the leg opposite the 30° angle.

Step 3: Calculate the Length of the Leg Opposite 30°

Using the equation from the 30°-60°-90° triangle properties:\[ \sqrt{3}x = 55 \]Solve for \( x \):\[ x = \frac{55}{\sqrt{3}} \]To express this without a square root in the denominator, multiply the numerator and denominator by \( \sqrt{3} \):\[ x = \frac{55 \sqrt{3}}{3} \] Approximation:\[ x \approx \frac{55 \times 1.732}{3} \approx 31.81\] mm.

Step 4: Calculate the Length of the Hypotenuse

The hypotenuse (\( 2x \)) in the triangle relates to the leg opposite the 30° angle as follows:\[ 2x = 2 \times \frac{55 \sqrt{3}}{3} = \frac{110 \sqrt{3}}{3} \] Approximation:\[ 2x \approx \frac{110 \times 1.732}{3} \approx 63.62 \] mm.

Key Concepts

Right Triangle

A right triangle is a triangle where one of its angles is exactly 90 degrees. This creates a special relationship between the sides of the triangle. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are known as the legs. In any right triangle, the Pythagorean theorem applies, which states that the sum of the squares of the two legs equals the square of the hypotenuse. This principle underpins why certain triangles, like the 30-60-90 triangle we're discussing, have predictable side ratios.
A key feature of right triangles is that one angle's measure ensures specific other angles if the triangle adheres to known types like the 30-60-90 triangle. Recognizing these helps solve for unknown sides or angles quickly.

Triangle Properties

Understanding the properties of triangles helps solve many geometric problems. A triangle is defined by its three sides and three angles. In any triangle, the sum of its interior angles is exactly 180 degrees. For the special 30°-60°-90° triangle, knowing one angle’s measure gives insight into the entire triangle.
In a 30°-60°-90° triangle:

• The angle measures are fixed at 30 degrees, 60 degrees, and 90 degrees.
• The side lengths are proportional, allowing for quick calculations and easy recognition. This is something called a reference triangle in geometry because of its simple, fixed ratio.

This proportion simplifies finding missing side lengths when at least one side is known, allowing us to leverage these properties for efficient calculations.

Ratio of Sides

The 30°-60°-90° triangle has sides that always conform to the ratio of 1: the square root of 3 :2. This means:

• The side opposite the 30° angle, known as the shortest side, is represented by 1.
• The side opposite the 60° angle, which is longer, is represented as the square root of 3 times the shortest side.
• The hypotenuse, opposite the 90° angle, is two times the shortest side.

Understanding this ratio is crucial because it allows you to solve related problems quickly. Given any one side, you can determine the others by applying these ratios. For instance, if the side opposite 60° is 55 mm, you can easily find the other sides using this established proportion, making it an incredibly helpful tool in geometry.