Problem 1 - Elementary and Intermediate Algebra

Question

Complete sentence with the best choice from the column on the right. In any ____ triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs. a) Hypotenuse b) Isosceles c) Legs d) Right e) Square roots f) \(30^{\circ}-60^{\circ}-90^{\circ}\)

Step-by-Step Solution

Answer

Right

Step 1 - Understanding the Sentence

Examine the provided sentence: 'In any ____ triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs.' Identify what type of triangle and what property is being described.

Step 2 - Identify the Property

The sentence refers to the property of a triangle where 'the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs.' This is known as the Pythagorean theorem.

Step 3 - Connect the Property to the Triangle Type

Determine which type of triangle the Pythagorean theorem applies to. The Pythagorean theorem is specific to right triangles.

Step 4 - Choose the Appropriate Option

Select the word 'right' from the options to complete the sentence accurately: 'In any **right** triangle, the square of the length of the hypotenuse is the sum of the squares of the lengths of the legs.'

Key Concepts

right triangle

A right triangle is a type of triangle that has one angle equal to 90 degrees. This right angle makes it special and gives it unique properties. The sides of a right triangle have specific names that help identify its parts.

These parts include:

• The two shorter sides are called the legs.
• The longest side, opposite the right angle, is called the hypotenuse.

Right triangles follow a special relationship known as the Pythagorean theorem. This theorem explains how the lengths of the sides are related.

hypotenuse

The hypotenuse is the longest side of a right triangle. It is always found opposite the right angle. Because it's the longest side, it has a special role in the Pythagorean theorem.

According to the theorem:

\[ c^2 = a^2 + b^2 \]

Here, \( c \) represents the length of the hypotenuse, while \( a \) and \( b \) represent the lengths of the two legs.

Understanding the hypotenuse is crucial since its length can be calculated if you know the lengths of the legs.

legs of a triangle

The legs of a triangle are the two sides that form the right angle in a right triangle. These sides play a key role in understanding and applying the Pythagorean theorem.

When using the theorem, the legs are represented as \( a \) and \( b \). The relationship between the legs and the hypotenuse is:

\[ a^2 + b^2 = c^2 \]

Breaking it down:

• \