Problem 20 - Prealgebra

Question

The lengths of three sides of a triangle are given. Determine whether each triangle is a right triangle. $$a=5, b=8, c=9$$

Step-by-Step Solution

Answer

The triangle is not a right triangle.

Step 1: Identify longest side

First, identify the longest side of the triangle, as it potentially represents the hypotenuse in a right triangle. The given sides are \(a = 5\), \(b = 8\), and \(c = 9\). Here, \(c = 9\) is the longest side.

Step 2: Apply the Pythagorean Theorem

To determine whether the triangle is a right triangle, use the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side (hypotenuse). This can be expressed as \(a^2 + b^2 = c^2\).

Step 3: Calculate the Square of Each Side

Calculate the square of each side: \(a^2 = 5^2 = 25\), \(b^2 = 8^2 = 64\), and \(c^2 = 9^2 = 81\).

Step 4: Verify the Pythagorean Condition

Now, check if the sum of the squares of the shorter sides equals the square of the longest side: \(25 + 64 = 89\). Since \(89 eq 81\), the Pythagorean condition is not satisfied.

Step 5: Conclude the Analysis

Since \(a^2 + b^2 eq c^2\), the triangle with sides 5, 8, and 9 is not a right triangle.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean Theorem is an essential concept in geometry, especially when dealing with right triangles. A right triangle is a triangle that has one angle measuring exactly 90 degrees. In such triangles, there is a special relationship between the lengths of the sides known as the Pythagorean Theorem. It states that if a triangle is a right triangle, then the sum of the squares of the two shorter sides (called legs) is equal to the square of the longest side (called the hypotenuse). The formula is expressed as:

• \( a^2 + b^2 = c^2 \)

In this formula:

• \( a \) and \( b \) are the lengths of the legs
• \( c \) is the length of the hypotenuse

Understanding this theorem helps in determining whether a given set of side lengths can form a right triangle by simply plugging in the side lengths into the formula and verifying if the equation holds true.

Examining the Triangle Inequality

The Triangle Inequality Theorem is another critical concept when dealing with triangles. It focuses on the relationship between the lengths of the sides of any triangle, not just right triangles. The theorem states that for a set of three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Mathematically, it's expressed as:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

Applying the Triangle Inequality helps ensure you aren't wasting time checking for a right triangle when the lengths can't even form any triangle. In our example, verifying these conditions with lengths 5, 8, and 9 ensures that they can indeed form a triangle, allowing us to proceed with further analysis like checking for right triangle characteristics.

Exploring Side Lengths of Triangles

When analyzing triangles, understanding the side lengths is crucial. Side lengths determine not only the type (like right, acute, or obtuse) but also whether the side lengths can form a triangle at all, based on the Triangle Inequality. In a mathematical problem like the one at hand, side lengths such as 5, 8, and 9 need to be critically examined.

• The longest side, here 9, is critical because in a right triangle, it's the hypotenuse, which means it needs to satisfy the Pythagorean Theorem with the other two sides.
• Calculating the square of each side is a key step, helping to explicitly demonstrate how they interact according to the theorem.

Through these calculations and comparisons, we determine the nature of the triangle mathematically, ensuring every step corroborates with each theorem discussed.