Question

Compare and contrast the \(30^{\circ}-60^{\circ}-90^{\circ}\) Triangle Theorem and the \(45^{\circ}-45^{\circ}-90^{\circ}\) Triangle Theorem.

Short answer

30°-60°-90° triangle has side ratio 1:√3:2; 45°-45°-90° triangle has ratio 1:1:√2.

Step-by-step solution

Understanding the Triangle Theorems

First, we need to understand the two special right triangles:1. The \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has angles of \(30^{\circ}, 60^{\circ}, \text{and } 90^{\circ}\). In this type of triangle, the side lengths are in the ratio \(1 : \sqrt{3} : 2\).2. The \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle has angles of \(45^{\circ}, 45^{\circ}, \text{and } 90^{\circ}\). This triangle has side lengths in the ratio \(1 : 1 : \sqrt{2}\).

Exploring the 30°-60°-90° Triangle

For the \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle:- The shortest side, opposite the \(30^{\circ}\) angle, is the reference side and has a length 'x'.- The side opposite the \(60^{\circ}\) angle is \(x\sqrt{3}\).- The hypotenuse, opposite the \(90^{\circ}\) angle, is \(2x\).

Examining the 45°-45°-90° Triangle

For the \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle:- Since this triangle is isosceles, the two legs are of equal length, each 'y'.- The hypotenuse is \(y\sqrt{2}\), opposite the \(90^{\circ}\) angle.

Comparing the Ratio of Sides

When comparing the two triangles:- The \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has sides in the ratio \(1 : \sqrt{3} : 2\).- The \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle has sides in the ratio \(1 : 1 : \sqrt{2}\).

Contrasting the Angles

Considering the angles:- The \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle has three different angles, leading to different side lengths.- The \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle has two equal angles, resulting in two equal side lengths (isosceles).

Key concepts

30-60-90 Triangle

The 30-60-90 triangle is one of the most significant special right triangles in geometry. This triangle has angles measuring 30 degrees, 60 degrees, and 90 degrees, making it highly recognizable. What's unique about the 30-60-90 triangle is its side length ratio, which is 1 : \( \sqrt{3} \) : 2. Here's what that means:
- Opposite the 30-degree angle, we have the shortest side, often considered the "base" or reference, denoted as \( x \).
- Opposite the 60-degree angle, the side length stretches to \( x\sqrt{3} \).
- Opposite the 90-degree angle is the hypotenuse, which measures \( 2x \).
This triangle ratio is a consistent pattern in all such triangles, making calculations and problem-solving more straightforward by knowing this simple ratio.
Being adept at identifying 30-60-90 triangles helps in many areas of math, including trigonometry and geometry problems.

45-45-90 Triangle

The 45-45-90 triangle is another important special right triangle, known for its symmetry. Both non-right angles in this triangle are 45 degrees. This makes it an isosceles right triangle, which means two of its sides are of equal length. The side length ratio for a 45-45-90 triangle is 1 : 1 : \( \sqrt{2} \).
- The two legs, which are opposite the 45-degree angles, have the same length, which can be denoted as \( y \).
- The hypotenuse, opposite the 90-degree angle, is \( y\sqrt{2} \).
This ratio helps quickly determine the lengths of any such triangle's sides when you know the length of one of its legs.
Since the triangle is isosceles by nature, it naturally shows the fundamental principles of equality of sides and angles, being an insightful example for discussing triangular properties.

Triangle Ratios

Understanding triangle ratios is crucial when dealing with special right triangles. These ratios help us to easily calculate unknown side lengths based on provided measurements. In a 30-60-90 triangle, as discussed, the ratio is 1 : \( \sqrt{3} \) : 2, whereas, in the 45-45-90 triangle, the ratio is 1 : 1 : \( \sqrt{2} \).
These ratios are not arbitrary but stem from mathematical principles that ensure each type of special triangle maintains consistent properties.

• Ratios allow for swift problem-solving, ensuring accuracy and saving time.
• They enable the solving of geometric problems even when direct measurements aren't available.
• Understanding and applying these ratios can form the basis for more complex trigonometric functions and calculations.

Mastering these ratios aids in many mathematical contexts, giving a deeper understanding of how angles relate to side lengths in right triangles.