Problem 24 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$a, b ; \quad c$$

Step-by-Step Solution

Answer

\( c = \sqrt{a^2 + b^2} \)

Step 1: Identify Pythagorean Theorem

In a right-angled triangle, the Pythagorean theorem relates the lengths of the sides. It states: For a triangle with right angle at \( \gamma = 90^\circ \), the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \). Mathematically, it is expressed as: \[ c^2 = a^2 + b^2 \]

Step 2: Solve for the Hypotenuse

Since we need to express the hypotenuse \( c \) in terms of the other two parts \( a \) and \( b \), we must rearrange the Pythagorean theorem. Take the square root of both sides to solve for \( c \):\[ c = \sqrt{a^2 + b^2} \]

Key Concepts

Understanding Right-Angled Triangles

A right-angled triangle is one of the most fundamental shapes in geometry. It is a type of triangle with one angle measuring exactly 90 degrees. This specific angle is referred to as the right angle. Having one angle fixed at 90 degrees gives right-angled triangles their unique properties and significance.
Perhaps the most interesting feature of a right-angled triangle is how it supports the Pythagorean theorem. The sides of a right-angled triangle have specific names:

• The hypotenuse: This is the side opposite the right angle and is the longest side of the triangle.
• The other two sides are commonly referred to as the 'legs' of the triangle.

Understanding these components and their relationships is crucial for constructing right-angled triangles and for solving exercises related to them.

The Hypotenuse: Central to the Pythagorean Theorem

The hypotenuse is an essential feature of every right-angled triangle. Its role is critical because it is the side directly opposite the right angle. Among the three sides of a right-angled triangle, the hypotenuse is always the longest.
In the context of the Pythagorean theorem, as seen in the exercise, the hypotenuse plays a pivotal role. According to the theorem, the length of the hypotenuse can be calculated using the formula:

• The square of the hypotenuse is equal to the sum of the squares of the other two sides.

This calculation provides a practical way to determine the hypotenuse when the lengths of the other two sides are known. Knowing how to find the hypotenuse is important for solving many geometry problems, making it a key concept to grasp.

Exploring the Mathematical Formula: Pythagorean Theorem

The Pythagorean theorem is a mathematical formula that defines a specific relationship in right-angled triangles. It states:

• The square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).

This relationship is expressed as: \[ c^2 = a^2 + b^2 \]When solving for the hypotenuse specifically, as in the exercise, this equation can be rearranged to solve for \( c \): \[ c = \sqrt{a^2 + b^2} \]Applying the theorem allows us to calculate unknown side lengths with ease, provided you have the lengths of the other two sides. It is a fundamental tool in mathematics that helps to seek solutions efficiently, making it vital for students to understand and be able to apply it correctly in exercises.