Problem 41 - Algebra 1: Concepts and Skills

Question

Determine whether the given lengths are sides of a right triangle. Explain your reasoning. $$ 3,9,10 $$

Step-by-Step Solution

Answer

No, the lengths 3, 9 and 10 cannot form a right triangle as they do not satisfy the Pythagorean theorem.

Step 1: Identify the Hypotenuse

First, identify the hypotenuse. In a right triangle, the hypotenuse is the longest side. In this case, it is of length 10.

Step 2: Apply the Pythagorean Theorem

Next, use the Pythagorean theorem, which in this case will take the form \(10^2 = 3^2 + 9^2\).

Step 3: Evaluate Each Side

Evaluate each side of equation. The left side, \(10^2\), equals 100. The right side, \(3^2 + 9^2\), equals 81+9=90.

Step 4: Compare the Results

As 100 is not equal to 90, the Pythagorean theorem does not hold, and the values 3, 9, 10 cannot be side lengths for a right triangle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one of its angles measuring 90 degrees, known as the right angle. This characteristic distinguishes it from other triangles and allows for specific mathematical properties to be applied. In a right triangle:

• The side opposite the right angle is called the hypotenuse and it is the longest side.
• The other two sides are commonly referred to as the legs.

Right triangles are fundamental in geometry and trigonometry because they serve as the basis for many mathematical theorems and formulas. Understanding the properties of right triangles can simplify the process of solving many real-world problems that involve angle and length measurements.

Hypotenuse

The hypotenuse is one of the critical components of a right triangle. As the longest side, it lies opposite the triangle's right angle. Identifying the hypotenuse is paramount when using the Pythagorean Theorem, as it helps in setting up the equation correctly. In any right triangle:

• The other two sides, or legs, connect to form the right angle.
• The length of the hypotenuse is always greater than the lengths of the other two sides.

In the example given, out of the sides 3, 9, and 10, the hypotenuse is 10, because it is the longest. Correctly identifying the hypotenuse ensures the accuracy of any calculations involved.

Triangle Sides

The three sides of a triangle are fundamental to its geometry. In the context of a right triangle, the relationship between these sides is especially important due to the Pythagorean Theorem. The sides of a triangle are:

• Hypotenuse: the longest side opposite the right angle.
• Legs: the two shorter sides that form the right angle.

It's crucial to measure the sides correctly to apply the Pythagorean Theorem accurately. In non-right triangles, the lengths of the sides do not hold the same special relationship. Therefore, using correct measurements and identifying which one is the hypotenuse allows for proper application of geometric principles.

Pythagorean Equation

The Pythagorean Equation is a powerful tool used to determine whether a set of three numbers can represent the side lengths of a right triangle. The theorem is stated as:\[ a^2 + b^2 = c^2 \]Where:

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

In the example provided, the equation becomes:\[ 3^2 + 9^2 = 10^2 \]This simplifies to\[ 9 + 81 = 100 \]While we find:\[ 90 eq 100 \]This tells us clearly that the numbers 3, 9, and 10 do not form a right triangle. The Pythagorean Equation, therefore, serves as a reliable check for the relationship between side lengths in right triangles.