Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 17-24: 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 Triangle Relationship

In any right triangle, the relationship between the sides is given by the Pythagorean theorem: \(c^2 = a^2 + b^2\). In triangle \(ABC\) with \(\gamma = 90^{\circ}\), \(c\) represents the hypotenuse.

Step 2: Express Hypotenuse in Terms of Legs

Given the relationship \(c^2 = a^2 + b^2\), solve for \(c\) by taking the square root of both sides yielding \(c = \sqrt{a^2 + b^2}\).

Step 3: Conclude Expression

Now that we've derived \(c = \sqrt{a^2 + b^2}\), we've expressed the hypotenuse \(c\) in terms of the legs \(a\) and \(b\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees, which is known as the right angle. This special characteristic means it has some unique properties:

• One of the sides is perpendicular to the base, forming the right angle.
• The side opposite the right angle is the largest side, called the hypotenuse.

In triangle ABC, where angle \( \gamma \) (often denoted as \( \angle C \)) is the right angle, the sides follow the rules laid out by the Pythagorean theorem. This theorem is specific to right triangles and helps us easily calculate the lengths of the sides when we know two of them.

Hypotenuse

The hypotenuse is the longest side in a right triangle and is opposite the right angle. In a right triangle, the hypotenuse has a crucial role because:

• It is always on one of the opposite corners from the right angle.
• It is the side that the Pythagorean theorem is primarily used to calculate.

When using the Pythagorean theorem, such as in triangle ABC, it states that the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides: \( c^2 = a^2 + b^2 \).
This equation shows the direct relationship between the hypotenuse and the legs (the other two sides), which helps in computational problems where you need to solve for unknown sides.

Triangle Sides

The sides of a right triangle include the two legs and the hypotenuse. Each has a specific role:

• Legs are the two sides that form the right angle.
• Their lengths are key contributors to determining the hypotenuse.

Using the Pythagorean theorem, you can express the hypotenuse \( c \) in terms of the legs \( a \) and \( b \):\[ c = \sqrt{a^2 + b^2} \]
This formula lets you find the hypotenuse if you know the lengths of the other two sides. Understanding how each side of the triangle relates to one another is essential, especially in trigonometry and geometry problems involving right triangles.