Problem 1 - Algebra 2 and Trigonometry

Question

Brandon said that if \(a\) is a positive real number, then \(3 a, 4 a,\) and 5\(a\) are the lengths of the sides of a right triangle. Therefore, \(3 \sqrt{2}, 4 \sqrt{2},\) and 5\(\sqrt{2}\) are the lengths of the sides of a right triangle. Do you agree with Brandon? Justify your answer.

Step-by-Step Solution

Answer

Yes, Brandon is correct. Both 3a, 4a, 5a and their scaled values satisfy the Pythagorean theorem.

Step 1: Understand the Triangle Condition

For three lengths to form the sides of a right triangle, they must satisfy the Pythagorean theorem. Specifically, if we have lengths \( c, b, \) and \( a \) such that \( c > b > a \), they form a right triangle if \( c^2 = a^2 + b^2 \).

Step 2: Check Pythagorean Relationship for 3a, 4a, 5a

Apply the Pythagorean theorem: \((5a)^2 = (3a)^2 + (4a)^2 \). Simplify each term: \((5a)^2 = 25a^2, (3a)^2 = 9a^2, (4a)^2 = 16a^2\). Add the results for \(3a\) and \(4a\): \(9a^2 + 16a^2 = 25a^2\). Since \(25a^2 = 25a^2\), they satisfy the theorem and indeed form a right triangle.

Step 3: Analyze Specific Case 3√2, 4√2, 5√2

Substitute \( a = \sqrt{2} \) into the proven form \(3a, 4a, 5a\). We have: \(3\sqrt{2}, 4\sqrt{2}, 5\sqrt{2}\). The factor \(a\) scales the sides of the right triangle, thus these values will also satisfy \((5\sqrt{2})^2 = (3\sqrt{2})^2 + (4\sqrt{2})^2\).

Step 4: Verify the Equalities

Calculate each side: \((5\sqrt{2})^2 = 25 \cdot 2 = 50\), \((3\sqrt{2})^2 = 9 \cdot 2 = 18\), and \((4\sqrt{2})^2 = 16 \cdot 2 = 32\). The sum for \(3\sqrt{2}\) and \(4\sqrt{2}\) becomes: \(18 + 32 = 50\). Since \(50 = 50\), this confirms they can be the sides of a right triangle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles measures exactly 90 degrees. This angle, known as the right angle, creates a distinct relationship between the three sides of the triangle.
The longest side opposite the right angle is called the hypotenuse, and the two other sides are referred to as the legs.

• One crucial property of a right triangle is the Pythagorean Theorem.
• This theorem states that the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)).

This is mathematically expressed as \(c^2 = a^2 + b^2\).
This foundational principle allows us to determine the validity of whether a set of three lengths can form a right triangle by checking for this specific equality.

Triangle Inequality

The triangle inequality theorem is an essential concept in understanding triangles, including right triangles. It posits that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For any triangle with sides \(a\), \(b\), and \(c\), the following inequalities hold:

• \[ a + b > c \]
• \[ b + c > a \]
• \[ a + c > b \]

These conditions ensure the formation of a proper triangle, preventing its sides from forming a straight line or collapsing into nothing. They are also crucial in determining the possible dimensions of triangles during the problem-solving process.
For right triangles, the triangle inequality helps reaffirm that any chosen sides must truly form a closed shape with the right angle.

Scaling Factors

Scaling factors play an intriguing role in the geometry of triangles, especially in affirming the Pythagorean Theorem for varied sizes. When each side of a triangle is multiplied by the same positive real number, the shape remains a triangle that is similar to the original.
The sides are proportionally increased or decreased, maintaining the angle measures and confirming the same triangle type.

• **Example Scenario**: Let the sides of a triangle be 3, 4, and 5. Scaling these sides by a factor \(a\) results in new side lengths: \(3a, 4a, 5a\).
• When scaling up or down by this factor, both the proportions of sides and the Pythagorean relationship hold true.

This is evident in the given exercise with the factor being \(\sqrt{2}\), proving that the scaled sides remain those of a right triangle. Therefore, scaling doesn't affect the essential properties of the triangle.

Real Numbers

Real numbers encompass a broad range of values, including whole numbers, fractions, and irrational numbers. They are fundamental in geometry and algebra.
Positive real numbers, in particular, are crucial in defining the lengths of sides in triangles, as they cannot be negative or imaginary.

• Considering any variable \(a\) as a positive real number, it guarantees that side lengths are positive, making them physically viable for a triangle.
• This includes square roots like \(\sqrt{2}\), which provide a precise scalar for scaling triangle sides.

The flexibility and continuity of real numbers allow triangles to take various forms while maintaining fundamental properties, as shown in the exercise where multiplication by \(\sqrt{2}\) yields real and practical side lengths that satisfy the conditions of a right triangle.