Problem 55 - Algebra and Trigonometry

Question

Show that a right triangle whose sides are in arithmetic progression is similar to a \(3-4-5\) triangle.

Step-by-Step Solution

Answer

A right triangle with sides in arithmetic progression is similar to a 3-4-5 triangle.

Step 1: Define the Sides in Arithmetic Progression

For a right triangle, the sides are denoted as \(a\), \(b\) (the legs), and \(c\) (the hypotenuse). For the sides to be in arithmetic progression, there is a common difference \(d\) such that \(b = a + d\) and \(c = a + 2d\). Now we have our sides in terms of \(a\) and \(d\).

Step 2: Apply the Pythagorean Theorem

Use the Pythagorean theorem, \(a^2 + b^2 = c^2\), for a right triangle:\[a^2 + (a + d)^2 = (a + 2d)^2\]Expanding both sides, we get:\[a^2 + (a^2 + 2ad + d^2) = a^2 + 4ad + 4d^2\]Simplifying gives:\[2a^2 + 2ad + d^2 = a^2 + 4ad + 4d^2\]

Step 3: Simplify the Equation

Continue simplifying the equation from Step 2:\[2a^2 + 2ad + d^2 = a^2 + 4ad + 4d^2\]Subtract \(a^2\) from both sides:\[a^2 + 2ad + d^2 = 4ad + 4d^2\]Rearrange terms: \[a^2 - 2ad - 3d^2 = 0\]

Step 4: Solve for \(a\) in terms of \(d\)

This is a quadratic equation in \(a\). Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -2d\), and \(c = -3d^2\):\[a = \frac{2d \pm \sqrt{(2d)^2 - 4 \cdot 1 \cdot (-3d^2)}}{2}\]\[a = \frac{2d \pm \sqrt{4d^2 + 12d^2}}{2}\]\[a = \frac{2d \pm 4d}{2}\]The simplified solutions are \(a = 3d\) or \(a = -d\). Since the side of a triangle cannot be negative, \(a = 3d\).

Step 5: Identify the Side Lengths of the Triangle

Given \(a = 3d\), substitute back to find \(b\) and \(c\):\[b = a + d = 3d + d = 4d\]\[c = a + 2d = 3d + 2d = 5d\]Thus, the side lengths in terms of \(d\) are \(3d, 4d,\) and \(5d\).

Step 6: Compare with 3-4-5 Triangle

The side lengths \(3d, 4d,\) and \(5d\) are proportional to the side lengths of the \(3-4-5\) triangle, since they are scalar multiples of 3, 4, and 5, respectively.

Key Concepts

Arithmetic Progression

In mathematics, an arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant. This difference is known as the common difference, denoted as \(d\). For example, the sequence \(2, 4, 6, 8\) has a common difference of 2.
In the problem of a right triangle with sides in an arithmetic progression, the sides must follow this sequence pattern. Consider sides \(a\), \(b\), and \(c\) such that \(b = a + d\) and \(c = a + 2d\). This ensures the sides increase in consistent steps, just as terms do in an arithmetic progression. Establishing this sequence is crucial as it sets the foundation for further proof steps using geometric properties.
Understanding arithmetic progression helps us determine the structure of the triangle's sides, which is essential for proving similarity to a 3-4-5 triangle. In this problem, knowing the sequence saves us from random guesses and guides us to logically find the side lengths consistent with geometric principles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation in Euclidean geometry that describes the relationship among the three sides of a right triangle. The theorem states that the square of the length of the hypotenuse \(c\) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\). This relationship can be mathematically expressed as:
\[ a^2 + b^2 = c^2 \]
This equation is central to solving problems involving right triangles, especially when the sides are not given explicitly. In the context of our problem, applying the Pythagorean Theorem allows us to confirm that the side lengths in an arithmetic progression indeed form a right triangle by satisfying this equation.
By substituting \(b\) and \(c\) with \(a + d\) and \(a + 2d\) into the theorem, we derived a quadratic equation in terms of \(a\) and \(d\). Successfully solving this helped us prove that the triangle sides are proportional to those in a 3-4-5 triangle. Without the Pythagorean Theorem, establishing the right-angle property and proving similarity would be much more challenging.

Quadratic Equation in Geometry

Quadratic equations frequently appear in geometric contexts where a relationship between factors like side lengths or angles is described. A standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Solving these equations often involves using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In the right triangle problem, after expanding the Pythagorean equation with arithmetic progression sides, we obtained a quadratic equation: \(a^2 - 2ad - 3d^2 = 0\). Solving it illuminates the relationship between side lengths \(a\), \(b\), and \(c\), and confirms the arithmetic progression. The solutions \(a = 3d\) or \(a = -d\) logically led us to use \(a = 3d\) because triangle sides cannot be negative.
Understanding quadratic equations and their solutions is important for students working through geometric problems. It shows how algebraic concepts can bridge to geometric proofs, and how solving for a variable helps determine actual measurements in practical contexts, such as determining if a triangle's sides are similar to a known ratio like 3-4-5.