Problem 25 - Elementary and Intermediate Algebra

Question

One leg of an isosceles right triangle is 3.2 feet long. Find the length of its hypotenuse. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

The hypotenuse is \( \sqrt{20.48} \) (exact) or approximately 4.52 feet.

Step 1: Understand the Problem

An isosceles right triangle has two equal sides, known as the legs, and a hypotenuse. We know the length of one leg, which is 3.2 feet. We need to find the length of the hypotenuse.

Step 2: Use the Pythagorean Theorem

For a right triangle, the Pythagorean theorem states: \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse. In an isosceles right triangle, \( a = b \). So, the formula simplifies to \( 2a^2 = c^2 \).

Step 3: Plug in the Given Value

Substitute the given length of the leg (3.2 feet) into the simplified formula: \( 2(3.2)^2 = c^2 \).

Step 4: Calculate \( c^2 \)

Calculate \( (3.2)^2 = 10.24 \). Multiply by 2 to find \( c^2 \): \( 2 \times 10.24 = 20.48 \).

Step 5: Find \( c \)

Take the square root of \( 20.48 \) to find \( c \): \( c = \sqrt{20.48} \).

Step 6: Approximate the Answer

Use a calculator to find the square root of \( 20.48 \) to two decimal places: \( \sqrt{20.48} \approx 4.52 \).

Key Concepts

Isosceles Right Triangle

An isosceles right triangle is a special type of right triangle that has two sides of equal length. These sides are called the legs of the triangle, and they are opposite the angles that are each 45 degrees, making this triangle not only isosceles but also a 45-45-90 triangle. The third side, the hypotenuse, is the longest side and is always opposite the right angle. Because the two legs are equal, a fascinating property of isosceles right triangles is that when you apply the Pythagorean Theorem, it simplifies the calculation process, as we'll see in the next section.
Basic characteristics of isosceles right triangles include:

• Two angles measuring 45 degrees
• A right angle measuring 90 degrees
• Two equal legs
• Hypotenuse opposite the right angle

Understanding these properties makes it easier to apply mathematical principles to solve problems involving isosceles right triangles.

Hypotenuse Calculation

To find the hypotenuse of a right triangle, we use the famous Pythagorean Theorem. This theorem is expressed as: \[ a^2 + b^2 = c^2 \] For an isosceles right triangle, since both legs are equal, say they are both of length \( a \), the equation simplifies to:\[ 2a^2 = c^2 \]This simplification occurs because \( b = a \) and thus \( a^2 + a^2 = 2a^2 \).

Using the given problem, where one leg is 3.2 feet, you substitute \( a \) with 3.2:\[ 2(3.2)^2 = c^2 \]Calculate \((3.2)^2 \) which is 10.24 and then multiply by 2 to get 20.48, the value for \( c^2 \). The final step to find \( c \) is taking the square root of 20.48. As we've seen, by identifying this pattern in isosceles right triangles, solving for the hypotenuse using the Pythagorean Theorem becomes straightforward.

Square Root Approximation

Calculating the exact value of the hypotenuse often requires determining the square root of a number. For our calculation, we need \( \sqrt{20.48} \). While calculators can give a quick decimal approximation, understanding what square roots imply can deepen your math insight.

Here are a few key points about square roots and approximations:

• The square root of a number \( x \) is a number \( y \) such that \( y^2 = x \).
• Square roots are not always whole numbers. Therefore, approximations help us express them in a more usable form.
• In this problem, \( \sqrt{20.48} \) approximates to 4.52. This tells us that repeating 4.52, squared, gets you close to 20.48.

Understanding how to approximate square roots is essential in geometry and when working with irrational numbers that appear often in calculating triangle dimensions. Always keep a calculator handy for these tasks, especially when a precise decimal is necessary.