Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. In any right triangle, the square of the hypotenuse is equal to the _____ of the squares of the two ____.

Step-by-Step Solution

Answer

sum; sides

Step 1: Identify the Theorem

This exercise refers to a fundamental concept in geometry, specifically the properties of right triangles. Recall that the relationship described involves a right triangle and is known as the Pythagorean Theorem.

Step 2: Recall the Pythagorean Theorem

The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, which are known as the legs.

Step 3: Replace Words Into the Blanks

From the Pythagorean Theorem, replace the appropriate words into the sentence. Thus, the square of the hypotenuse is equal to the 'sum' of the squares of the two 'legs' or 'sides'.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one of its angles precisely equal to 90 degrees. This is known as the right angle. The other two angles must add up to 90 degrees because the sum of all angles in any triangle is always 180 degrees. In a right triangle, there are specific terms used to describe its sides:

• The side opposite the right angle is called the hypotenuse, which is typically the longest side.
• The other two sides, which form the right angle, are known as the legs or catheti.

Right triangles have unique properties and relationships between their sides and angles, making them an essential topic in geometry. Understanding these properties helps in solving various mathematical problems, especially those involving measurements and distances.

Hypotenuse

The hypotenuse is a critical element in any right triangle. It is the side that directly faces the right angle. Being the longest side, the hypotenuse plays a central role in the Pythagorean Theorem.
The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Mathematically, it is represented as:
\[ c^2 = a^2 + b^2 \]
where:

• \(c\) is the hypotenuse.
• \(a\) and \(b\) are the lengths of the other two sides.

This algebraic relationship is foundational in geometry, allowing for the determination of one side if the other two are known. Recognizing the hypotenuse in a right triangle is essential for properly applying the Pythagorean Theorem in problem-solving scenarios.

Geometry

Geometry is the branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. Understanding right triangles and the Pythagorean Theorem is a fundamental part of geometry. This theorem is not only crucial for solving right triangle problems but is also applied in various real-world situations, like architecture and engineering.
In geometry, problems often involve calculating lengths and areas using formulas derived from the properties of geometric shapes. When working with right triangles in particular, the Pythagorean Theorem allows us to:

• Determine unknown side lengths in right triangles.
• Verify whether a set of three side lengths can form a right triangle.
• Solve various practical problems related to space and distance.

By recognizing the relationships and applying theorems like the Pythagorean Theorem, we can better understand how different geometric principles come together in a coherent way, adding to our overall comprehension of mathematics.