Problem 93 - Elementary Algebra

Question

If the two legs of a right triangle measure 9 units and 12 units, then find the length of the hypotenuse.

Step-by-Step Solution

Answer

The hypotenuse is 15 units long.

Step 1: Identify the Formula

To find the length of the hypotenuse in a right triangle, we use the Pythagorean theorem. The theorem is stated as: \[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides, or legs.

Step 2: Plug in the Known Values

We are given that the two legs of the triangle have lengths 9 units and 12 units. Let's assign:\[ a = 9, \, b = 12 \]We substitute these values into the Pythagorean theorem:\[ c^2 = 9^2 + 12^2 \]

Step 3: Calculate the Squares

We calculate the squares of the given leg lengths:\[ 9^2 = 81 \]\[ 12^2 = 144 \]

Step 4: Sum the Squares

Add the squares from the previous step to find \( c^2 \):\[ c^2 = 81 + 144 = 225 \]

Step 5: Solve for the Hypotenuse

To find \( c \), take the square root of 225:\[ c = \sqrt{225} = 15 \]

Key Concepts

Right Triangle

In geometry, a right triangle is a special type of triangle which has one angle that is exactly 90 degrees. This is known as the right angle. The side opposite this right angle is the longest side of the triangle and it is called the hypotenuse. The other two sides are referred to as the legs of the triangle.

Right triangles follow a unique property described by the Pythagorean Theorem, which relates the lengths of the three sides in a straightforward way. Understanding the nature of right triangles is essential in solving many geometry problems, especially when it comes to calculating distances or crafting constructions. Knowing that one angle is 90 degrees simplifies many calculations and provides a foundational concept in trigonometry.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. Whenever you need to calculate its length, you can rely on the Pythagorean theorem:

• The hypotenuse is significant because it is the side that connects distances in dimensional space.
• In practical terms, finding the hypotenuse can be seen as finding the most direct path across a rectangular space, like cutting across a field instead of walking around it.

By determining the length of the hypotenuse, we gather critical information about the triangle's size and shape. This concept is not only foundational in geometry but also plays an essential role in physics, engineering, and many fields where straight-line distances are calculated.

Mathematical Formula

Mathematics often uses formulas as compact ways to convey essential information and calculations. The formula for the Pythagorean theorem is a perfect example of such utility. It is expressed as:\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the right triangle.

• This formula allows for quick calculation of any one side when the other two sides are known, making it extremely versatile.
• Understanding the underlying reasons why the formula works involves appreciating how areas of squares on the triangle's sides relate to each other geometrically.

It is fundamental not only to geometry, but it also extends its usefulness beyond into algebra and even calculus, functioning as a bridge to understand deeper concepts.

Solving Geometry Problems

Geometry problems often require a systematic approach to identify the information needed and the formulas or theorems applicable. When solving problems related to right triangles, it is crucial to:

• Identify the right triangle and label its sides accordingly – hypotenuse always opposite the right angle.
• Use known side lengths to apply the Pythagorean theorem, substituting the values into the correct places in the formula.
• Verify the solution through logical checks, ensuring all mathematical operations like squares and square roots are accurately executed.

By practicing the methodical application of the Pythagorean theorem in various scenarios, one can develop a strong ability to solve geometry problems. It not only aids in academic settings but also enhances spatial reasoning skills for practical everyday problem-solving.