Problem 51 - Precalculus: Mathematical for Calculus

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

The triangle is similar to a 3-4-5 triangle because its side ratios match 3:4:5.

Step 1: Understand the Concept of Arithmetic Progression

In an arithmetic progression, the difference between consecutive terms is constant. Let's denote the sides of the triangle as \(a\), \(a+d\), and \(a+2d\), where \(d\) is the common difference. This setup assumes \(a\), \(a+d\), and \(a+2d\) are in increasing order.

Step 2: Apply the Pythagorean Theorem

For a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. Thus, we apply the Pythagorean theorem: \((a+2d)^2 = a^2 + (a+d)^2\).

Step 3: Simplify the Pythagorean Equation

Expand and simplify the equation: \(a^2 + 4ad + 4d^2 = a^2 + a^2 + 2ad + d^2\), which reduces to \(4ad + 4d^2 = a^2 + 2ad + d^2\).

Step 4: Rearrange and Solve the Quadratic Equation

Rearrange terms to obtain\(0 = a^2 - 2ad - 3d^2\). This is a quadratic equation in \(a\), which can be solved using the quadratic formula: \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(b = -2d\) and \(c = -3d^2\).

Step 5: Calculate the Values of 'a'

Substitute \(b = -2d\) and \(c = -3d^2\) into the quadratic formula. Solving \(a^2 - 2ad - 3d^2 = 0\) gives \(a = 3d\).

Step 6: Verify Similarity to a 3-4-5 Triangle

Substitute \(a = 3d\) back to the sides of the triangle, giving sides \(3d, 4d, 5d\). The ratios of sides \(3:4:5\) match the sides of a 3-4-5 triangle, confirming that the triangle is similar to a 3-4-5 triangle.

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence where the difference between consecutive terms remains constant. Imagine this as a series of numbers lined up in such a way that each number, apart from the first, can be determined by adding a constant value to the preceding number. In the context of our right triangle problem, we label the sides as \(a\), \(a+d\), and \(a+2d\), with \(d\) being the common difference. This formulation is essential as it helps in setting a base framework to tackle this geometric problem effectively.

• First Side: \(a\)
• Second Side: \(a + d\)
• Third Side: \(a + 2d\)

In such a triangle, each side increases systematically, making arithmetic progression a key concept to uncover the triangle's properties.

Pythagorean Theorem

The Pythagorean theorem is a foundational principle in geometry, particularly dealing with right triangles. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This relation is not merely a formula but a crucial rule to explore and validate a triangle's right-angle nature.
Apply this principle to our arithmetic progression triangle: \((a+2d)^2 = a^2 + (a+d)^2\).

• The square of the longest side, \((a + 2d)^2\), is equivalent to the square of side \(a\) plus the square of side \(a + d\).

Using the Pythagorean theorem here helps to verify the side association and check for similarity to a known triangle, like the 3-4-5 triangle.

Similar Triangles

Triangles are called similar if they have the same shape, but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are in proportion. For our triangle, we need to prove it is similar to the 3-4-5 triangle by showing that the side ratios are equivalent.
Once calculated, the sides of our triangle appear as:

• \(3d\), \(4d\), \(5d\)

The side ratios \(3:4:5\) clearly correspond to the renowned 3-4-5 triangle. Therefore, knowing about similar triangles helps immensely in understanding the proportional relationships of different geometric figures.

Quadratic Equation

Quadratic equations are a staple in solving real-world problems, including those in geometry. They are polynomial equations of the second degree and can often reveal crucial numeric relationships. The general form is recorded as \(ax^2 + bx + c = 0\).
In the exercise, we derive the quadratic equation:

• \(a^2 - 2ad - 3d^2 = 0\)

This arises while equating and analyzing our right triangle sides in arithmetic progression. Solving this using the quadratic formula provides the value of \(a\) as \(3d\), a step that reinforces our access to validate similarities with the 3-4-5 triangle. Mastery of quadratic equations thus acts as a powerful tool to unearth these geometric relationships efficiently.